Solution: In 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62

The ____ intercepts of the equation 9x2 + 4y = 36 are 3. To graph y = x2 - 4, you would shift the graph of y = x2 . (pp. 1 65-] 66)

True or False The expression 4x3 - 3.6x2 - V2 is a polynomial. (pp. 39--47

To graph y = x2 - 4, you would shift the graph of y = x2 _____ a distance of units _____. (pp. 252-260)

True or False The x-intercepts of the graph of a function y = f(x) are the real solutions of the equation f(x) = O. (pp. 224-225 )

The graph of every polynomial function is both ____ and ____.

A real number r for which fer) = 0 is called a(n) ____ of the function f.

If r is a real zero of even multiplicity of a function f, the graph of f the x-axis at r.

True or False The graph of f(x) = x2(x - 3) (x + 4) has exactly three x-intercepts.

True or False The x-intercepts of the graph of a polynomial function are called turning points.

True or False End behavior: the graph of the function f(x) = 3x 4 - 6x2 + 2x + 5 resembles y = X4 for large values of Ixl . yes

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = 4x + x3

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = 5x2 + 4X4

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. g(x) = 2

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. hex) = 3 - 2x

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f (x) = 1 - - x

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = x(x - 1)

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. g(x) = X?/2 - x2 + 2

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. hex) = vx(vx - 1)

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. F(x) = 5x - 7rX' + 2

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. F(x) = -, x-

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. G(x) = 2(x - 1 )2(x2 + 1)

In Problems 11-22, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. G(x) = -3x2 (x + 2)3

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = (x + 1 )4

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = (x - 2)5

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = x5 - 3

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f ( x) = X4 + 2

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = 2X4

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = 3xs

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = -x5

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = -x4

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = (x - 1 )5 + 2

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = (x + 2 )4 - 3

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = 2 (x + 1 )4 + 1

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = - (x - 1 )) - 2

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = 4 - (x - 2)5

In Problems 23-36, use transformations of the graph of y = X4 or y = x5 to graph each function. f(x) = 3 - (x + 2)4

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -1, 1,3; degree 3

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -2, 2,3; degree 3

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -3, 0, 4; degree 3

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -4, 0, 2; degree 3

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -4, -1, 2, 3; degree 4

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -3, -1, 2, 5; degree 4

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -1, multiplicity 1; 3, multiplicity 2; degree 3

In Problems 37-44, form a polynomial whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient. Zeros: -2, multiplicity 2; 4, mUltiplicity 1; degree 3

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = 3(x - 7)(x + 3)2

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = 4(x + 4)(x + 3)3

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = 4(x2 + l)(x - 2)3

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = 2 (x - 3)(x + 4)3

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = -2 (x + 1)2 (x2 + 4)2

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = (x - y(X - 1 )3

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = (x - 5)3(x + 4)2

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = (x + V3/(x - 2 ) 4

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = 3(x2 + 8)(x2 + 9)2

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = -2 (x2 + 3)3

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = -2x2 (x2 -2)

In Problems 45-56, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph off resembles for large values of Ixl . f(x) = 4x(x2 - 3)

In Problems 57-60, identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not. 4 4 x

In Problems 57-60, identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not. -4 -2 -2 -4 2 4 x

In Problems 57-60, identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not. -2 x

In Problems 57-60, identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not. -4 - 2 -2 2 4 x

In Problems 61-64, decide which of the polynomial functions in the list might have the given graph. (More than one answer may be possible.) Y x (a) y = -4x(x - l)(x - 2) (b) y = x2(x - 1 )2(x - 2) (c) y = 3x(x - 1)(x - 2) (d) y = x(x - 1 )2(x - 2? (e) y = x3(x - 1)(x - 2) (f) Y = -x(1 - x)(x - 2)

In Problems 61-64, decide which of the polynomial functions in the list might have the given graph. (More than one answer may be possible.) r I .o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x)

In Problems 61-64, decide which of the polynomial functions in the list might have the given graph. (More than one answer may be possible.) Y 2 -2 1 ? (a) y = - (x- - 1)(x - 2) 2 1 (b) y = --(x2 + 1)(x - 2) 2 (c) Y = (x2 - 1 ) ( 1 - ) I? ? (d) y = - Z (x- 1 )-(x - 2) 3 x (e) y = (x2 + D(x2 - 1)(2 - x) (f) y = - (x - 1)(x - 2 ) (x + 1)

In Problems 61-64, decide which of the polynomial functions in the list might have the given graph. (More than one answer may be possible.) Y 2 -2 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 1 )2

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 2)3

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x2(x - 3)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x(x + 2 )2

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = 6x3(x + 4)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = 5x(x - 1 )3

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = -4x2(x + 2 )

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = - x\x + 4)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - l )(x - 2 ) (x + 4)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x + l)(x + 4) (x - 3)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = 4x - x3

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x - x3

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x2(x - 2 ) (x + 2 )

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x2(x - 3)(x + 4)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x + 2 ) 2(x - 2 )2

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x + 1 )3(x - 3)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 1 )2(x - 3) (x + 1 )

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x + 1 )2(x - 3) (x - 1)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x + 2?(x - 4?

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 2 )2(x + 2 ) (x + 4)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x2(x - 2) (x2 + 3)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x2(x2 + l )(x + 4)

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = -x2(X2 - l ) (x + 1 )

In Problems 65-88: (a) Find the x- and y-intercepts of each polynomial function f. 62. r I 64. o'pdx (a) y = 2x3(x - l)(x - 2)2 (b) y = x2(x - 1)(x - 2) (c) y = x3(x - 1 )2(x - 2) Y 2 -2 (d) y = x2(x - 1 )2(x - 2)2 (e) y = 5x(x - 1 )2(x - 2) (f) y = -2x(x - 1 )2(2 - x) 3 x 1 (a) y = - Z (x2 - 1)(x - 2)(x + 1) 1 (b) y = - Z (x2 + l)(x - 2)(x + 1) 1 ? (c) Y = - Z (x + l )- (x - l)(x - 2) (d) y = (x - 1?(x + 1)( 1 - ) (e) y = - (x - 1?(x - 2)(x + 1) (f) y = _ (X2 + D(x - 1?(x + 1)(x - 2) (b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept. (c) End behavior:find the power function that the graph of f resembles for large values of Ix!. (d) Determine the maximum number of turning points on the graph of f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = -x2(x2 - 4) (x - 5)

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = x3 + 0.2x2 - l.5876x - 0.31752

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = x3 - 0.8x2 - 4.6656x + 3.73248

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = x3 + 2.56x2 - 3.31x + 0.89

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = x3 - 2.91x2 - 7.668x - 3.8151

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = x4 - 2.5x2 + 0.5625

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = X4 - 1 8.5x2 + 50.2619

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = 2X4 - 7TX3 + V5x - 4

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = - 1.2x4 + 0.5x2 - v3x + 2

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = -2x5 - V2x2 - X - V2

In Problems 89-98, for each polynomial function f: 73. f(x) = (x - l )(x - 2 ) (x + 4) 76. f(x) = x - x3 79. f(x) = (x + 2 ) 2(x - 2 )2 82. f(x) = (x + 1 )2(x - 3) (x - 1) 85. f(x) = x2(x - 2) (x2 + 3) 88. f(x) = -x2(x2 - 4) (x - 5) (a) Find the degree of the polynomial. D etermine the end behavior: that is, find the power function that the graph of f resembles for large values of lxl . (b) Graph f using a graphing utility. (c) Find the x- and y-intercepts of the graph, rounded to two decimal places. (d) Use TABLE to find points on the graph around each x-intercept. Determine on which intervals the graph is above and below the x-axis. (e) Determine the local maxima and local minima, if any exist, rounded to two decimal places. That is, locate any turning points. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f by hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of I Use the graph to find the range of r (h) Use the graph to determine where f is increasing and where f is decreasing. f(x) = 7TX5 + 7TX4 + v3x + 1

Hurricanes In 2005, hurricane Katrina struck the Gulf Coast of the United States, killing 1289 people and causing an estimated $200 billion in damage. The following data represent the number of major hurricane strikes in the US. (category 3, 4, or 5) each decade from 1921 to 2000. (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) The cubic function of best fit to these data is (c) Use a graphing utility to verify that the function given in part (b) is the cubic function of best fit. (d) With a graphing utility, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram. (e) Concern has risen about the increase in the number and intensity of hurricanes, but some scientists believe this is just a natural fluctuation that could last another decade or two. Use your model to predict the number of major hurricanes that Source: National Oceanic & Atmospheric Administration will strike the United States between 2001 and 2010. Does your result appear to agree with what these scientists believe? (f) From 2001-2005, five major hurricanes had struck the United States. Does this support or contradict your prediction in part (e)?

Active Duty The following data represent the number of active duty military personnel (in millions) in the four major branches of the U.S. military for the years 1998-2005, where 1 represents 1998, 2 represents 1999, and so on. Year, x Active Duty Personnel, N 1998, 1 1 .41 1999, 2 1 .39 2000, 3 1.38 2001, 4 1.39 2002, 5 1.41 2003, 6 1.42 2004, 7 1.41 2005, 8 1.39 Source: infoplease.com (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) The cubic function of best fit to these data is N(x) = -0.00146x3 + 0.01 98x2 - 0.0742x + 1 .4671 Use this function to predict the number of active duty military personnel in 2002. , (c) Use a graphing utility to verify that the function given in part (b) is the cubic function of best fit. (d) With a graphing utility, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram. ( e) Do you think that the function given in part (b) will be useful in predicting the number of active duty military personnel in 2010? Explain.

Temperature The following data represent the temperature T (OFahrenheit) in Kansas City, Missouri, x hours after midnight on May 15, 2005. ours after Midnight, x 6 9 12 15 18 21 24 Temperature (OF), T 45.0 44.1 51.1 57.9 63.0 63.0 59.0 54.0 Source: The Weather Underground (a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) Find the average rate of change in temperature from 9 AM to 12 noon. (c) What is the average rate of change in temperature from 3 PM to 6 PM? (d) The cubic function of best fit to these data is T(x) = -0.0103x3 + 0.32x2 - 1.37x + 45.39 Use this function to predict the temperature at 5 PM. lfl (e) Use a graphing utility to verify that the function given in part (b) is the cubic function of best fit. (I') With a graphing utility, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram. (g) Interpret the y-intercept.

Can the graph of a polynomial function have no y-intercept? Can it have no x-intercepts? Explain.

Write a few paragraphs that provide a general strategy for graphing a polynomial function. Be sure to mention the following: degree, intercepts, end behavior, and turning points.

Make up a polynomial that has the following characteristics: crosses the x-axis at -1 and 4, touches the x-axis at 0 and 2, and is above the x-axis between 0 and 2. Give your polynomial to a fellow classmate and ask for a written critique of your polynomial.

Make up two polynomials, not of the same degree, with the following characteristics: crosses the x-axis at -2, touches the x-axis at 1, and is above the x-axis between -2 and 1. Give your polynomials to a fellow classmate and ask for a written critique of your polynomials.

The graph of a polynomial function is always smooth and continuous. Name a function studied earlier that is smooth and not continuous. Name one that is continuous, but not smooth.

Which of the following statements are true regarding the graph of the cubic polynomial [(x) = x3 + bx2 + ex + d? (Give reasons for your conclusions.) (a) It intersects the y-axis in one and only one point. (b) It intersects the x-axis in at most three points. (c) It intersects the x-axis at least once. (d) For Ixl very large, it behaves like the graph of y = x 3 . (e) It is symmetric with respect to the origin. (f) It passes through the origin.

The illustration shows the graph of a polynomial function. (a) Is the degree of the polynomial even or odd? (b) Is the leading coefficient positive or negative? (c) Is the function even, odd, or neither? y x (d) Why is x2 necessarily a factor of the polynomial? (e) What is the minimum degree of the polynomial? (f) Formulate five different polynomials whose graphs could look like the one shown. Compare yours to those of other students. What similarities do you see? What differences?

Design a polynomial function with the following characteristics: degree 6; four distinct real zeros, one of multiplicity 3;yintercept 3; behaves like y = -5x6 for large values of Ixl. Is thjs polynomial unique? Compare your polynomial with those of other students. What terms will be the same as everyone else's? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the polynomial?

True or False The quotient of two polynomial expressions is a rational expression. (pp. 61-68)

What is the quotient and remainder when 3x4 - x2 is divided by x3 - x2 + 1. (pp. 44-47)

Graph y = - . (pp. 1 70-171) x

Graph y = 2(x + 1)2 - 3 using transformations. (pp. 2S2-260)

The line __ is a horizontal asymptote of R x = --3 -' X + 1

The line __ is a vertical asymptote of R(x) = : :

For a rational function R, if the degree of the numerator is less than the degree of the denominator, then R is __ .

True or False The domain of every rational function is the set of all real numbers.

True or False If an asymptote is neither horizontal nor vertical, it is called oblique.

True or False If the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficients gives rise to the horizontal asymptote.

In Problems 11-22, find the domain of each rational function. R(x) = -- 12 R( ) x-3

In Problems 11-22, find the domain of each rational function. ( ) . x = 3+x

In Problems 11-22, find the domain of each rational function. H(x) = -4x2 (x - 2)(x + 4)

In Problems 11-22, find the domain of each rational function. G(x) = (x + 3) (4 - x)

In Problems 11-22, find the domain of each rational function. F ( x) = 2--'--- --'--- 2x - Sx - 3

In Problems 11-22, find the domain of each rational function. Q ( x) = - 3x --:- 2 - + - Sx - --'--- 2

In Problems 11-22, find the domain of each rational function. R(x) = - X 3 - 8

In Problems 11-22, find the domain of each rational function. R(x) = 4 x - I

In Problems 11-22, find the domain of each rational function. H(x) = 3x2 + X x2 + 4

In Problems 11-22, find the domain of each rational function. G(x) = X + 1

In Problems 11-22, find the domain of each rational function. R(x) = 2 ---'- 4(x - 9)

In Problems 11-22, find the domain of each rational function. F (x) = --'-----'- - 3(x2 + 4x + 4)

In Problems 23-28, use the graph shown to find: (a) The domain and range of each function (d) Vertical asymptotes, if any (c) Horizontal asymptotes, if any y 4 -4 -4 --l 4 x

In Problems 23-28, use the graph shown to find: (a) The domain and range of each function (d) Vertical asymptotes, if any (c) Horizontal asymptotes, if any -3 I I I I I I 1 -3

In Problems 23-28, use the graph shown to find: (a) The domain and range of each function (d) Vertical asymptotes, if any (c) Horizontal asymptotes, if any -3

In Problems 23-28, use the graph shown to find: (a) The domain and range of each function (d) Vertical asymptotes, if any (c) Horizontal asymptotes, if any 3 --l y 3 x

In Problems 23-28, use the graph shown to find: (a) The domain and range of each function (d) Vertical asymptotes, if any (c) Horizontal asymptotes, if any 3 I I I I I I I _ _ _ -.-l _ _ _ _ _ __ L __ _ -3 3 x

In Problems 23-28, use the graph shown to find: (a) The domain and range of each function (d) Vertical asymptotes, if any (c) Horizontal asymptotes, if any x

In Problems 29-40, graph each rational function using transformations. F(x) = 2 + 1

In Problems 29-40, graph each rational function using transformations. Q(x) = 3 + ? x

In Problems 29-40, graph each rational function using transformations. R(x) = (x - If

In Problems 29-40, graph each rational function using transformations. R(x) = - x

In Problems 29-40, graph each rational function using transformations. H ( x) = x + 1

In Problems 29-40, graph each rational function using transformations. G(x) = (x + 2f

In Problems 29-40, graph each rational function using transformations. R(x) = x2 + 4x + 4

In Problems 29-40, graph each rational function using transformations. R(x) = -- + 1 x-I

In Problems 29-40, graph each rational function using transformations. G(x) = 1 + (x _ 3) 2

In Problems 29-40, graph each rational function using transformations. F(x) = 2 - x + 1

In Problems 29-40, graph each rational function using transformations. R (x) = x- 4 x2

In Problems 29-40, graph each rational function using transformations. R(x) = x - 4 x

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. R(x) = x + 4

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. R(x) = 3x + 5 x - 6

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. H (x) = x 2 _ 5x + 6

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. G(x) = x2 _ 5x + 6

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. T(x) = x4 _ 1

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. P(x) = x3 _ 1

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. Q(x) =

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. F(x) = 2x3 + 4x2

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. R(x) = x3 + 3x

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. R(x) = 6x2 + X + 12 3x2 - 5x - 2

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. G ( x) = x3 - 1 x - x2

In Problems 41-52, find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. F(x) = x _ x3

Gravity In physics, it is established that the acceleration due to gravity, g (in meters/sec2 ), at a height h meters above sea level is given by 3.99 X 101 4 g(h) = (6.374 X 106 + h )2 where 6.374 X 106 is the radius of Earth in meters. (a) What is the acceleration due to gravity at sea level? (b) The Sears Tower in Chicago, Illinois, is 443 meters tall. What is the acceleration due to gravity at the top of the Sears Tower? (c) The peak of Mount Everest is 8848 meters above sea level. What is the acceleration due to gravity on the peak of Mount Everest? (d) Find the horizontal asymptote of g( h). (e) Solve g(h) = O. How do you interpret your answer?

Population Model A rare species of insect was discovered in the Amazon Rain Forest. To protect the species, environmentalists declare the insect endangered and transplant the insect into a protected area. The population P of the insect ( months after being transplanted is 50( 1 + 0.5t) pe t) = (2 + O.Olt) (a) How many insects were discovered? In other words, what was the population when t = O? (b) What will the population be after 5 years? (c) Determine the horizontal asymptote of pet). What is the largest population that the protected area can sustain?

R esistance in Parallel Circuits From Ohm's law for circuits, it follows that the total resistance Rtot of two components hooked in parallel is given by the equation RIR R =--- 2 tot RI + R 2 where Rl and R 2 are the individual resistances. (a) Let Rl = 10 ohms, and graph Rtot as a function of R 2. (b) Find and interpret any asymptotes of the graph obtained in part (a). (c) If R 2 = 2, what value of RI will yield an Rtot of 1 7 ohms? Source: en. wikipedia.org/wiki/SeriesJmd_paralleCcircuits

Newton's Method In calculus you will learn that, if p(x) = a"x " + a, ,_lx, , -I + ... + ajX + ao 3. is a polynomial, then the derivative of p(x) is p'(x) = na"x , ,-I + (n - 1)a, ,_jx, , -2 + . . . + 2a2 x + al Newton's Method is an efficient method for finding the x-intercepts (or real zeros) of a function, such as p(x). The steps below outline Newton's Method STEP 1: Select an initial value Xo that is somewhat close to the x-intercept being sought. STEP 2: Find values for x using the relation p(X, ,) X, ,+ I = X" - -(-) n = 1, 2, ... p x" until you get two consecutive values Xk and Xk+1 that agree to whatever decimal place accuracy you desire. STEP 3: The approximate zero will be Xk+l ' Consider the polynomial p(x) = x3 - 7x - 40. (a) Evaluate p(5) and p( -3). (b) What might we conclude about a zero of p. Explain. (c) Use Newton's Method to approximate an x-intercept, 1', -3 < r < 5, of p(x) to four decimal places. .- (d) Use a graphing utility to graph p(x) and verify your answer in part (c). (e) Using a graphing utility, evaluate per) to verify your result.

If the graph of a rational function R has the vertical asymptote x = 4, the factor x - 4 must be present in the denominator of R. Explain why.

If the graph of a rational function R has the horizontal asymptote y = 2, the degree of the numerator of R equals the degree of the denominator of R. Explain why.

Can the graph of a rational function have both a horizontal and an oblique asymptote? Explain.

Make up a rational function that has y = 2x + 1 as an oblique asymptote. Explain the methodology that you used.

The intercepts of the equation y = -- are ____ . (pp. 1 65-1 66)

If the numerator and the denominator of a rational function have no common factors, the rational function is ___ _

True or False The graph of a polynomial function sometimes has a hole.

True or False The graph of a rational function never intersects a horizontal asymptote.

True or False The graph of a rational function sometimes intersects a vertical asymptote.

True or False The graph of a rational function sometimes has a hole.

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = x( v + 4)

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = A ( x - 1)( x+ 2)

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = 3x + 3 2x + 4

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = 2x + 4 x-I

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = -- X - 4

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = -?:--- r - x - 6

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. P( x) = ---: 2 - 1

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. Q(x) = ? x- 4

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. H(x) = -- r - 9

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. G(x) = x- + 2x

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = x2 x2 + x - 6

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = x2 + X - 12 x2 - 4

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. G(x) = - 2 - X - 4

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. G(x) = -- x- - 1

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = ? (x - 1)(r - 4)

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R (x) = ----- (x + 1) (x2 - 9)

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. H (x) = X - 16

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. H(x) = x2 + 4 x4 - 1 X - 16

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. F(x) = --- x+2

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. F ( x) = x2 + 3x + 2 x-I

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = x2 + X - 12 x-4

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R x - _ x+)

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. F(x) = x2 + X - 12 x+2

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. G(x) = --- x + 1

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = , (x + 3)"

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = ? x(x - 4)-

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = x2 ? + X - 12 x- -x-6

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R x - ? r + 8x + 15

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = -----=--- 2r - 7x + 6

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R x - 2x 2 - x - 15

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R(x) = ,., x + j

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. R( ) = x2 + X - 30 3. x x + 6

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. f(x) = x + - x

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. f(x) = 2x + - x

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. f(x) = r + - x

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. f(x) = 2x2 + - X

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. f (x) = x + 3" x

In Problems 7-44, follow Steps 1 through 7 on page 355 to analyze the graph of each function. f(x) = 2x + 3" x

In Problems 45-48, find a rational function that might have the given graph. (More than one answer might be possible.) )[-': l '-: '1 ------- ---f- y=1 -3

In Problems 45-48, find a rational function that might have the given graph. (More than one answer might be possible.) '-: '1 3

In Problems 45-48, find a rational function that might have the given graph. (More than one answer might be possible.) -4 -3 -2 x=2

In Problems 45-48, find a rational function that might have the given graph. (More than one answer might be possible.) 15 -10 3 x= Y I 10 -5 [ 8 6 4 [ -2 [ [ -4 [ [ -6 [ -8 x x= 4 -------------y=3 [ [ [5 10 15 20 x

Drug Concentration The concentration C of a certain drug in a patient's bloodstream I hours after injection is given by 1 C(I) = 212 + 1 (a) Find the horizontal asymptote of C(I). What happens to the concentration of the drug as I increases? c, (b) Using your graphing utility, graph C(t). , (c) Determine the time at which the concentration is highest.

Drug Concentration The concentration C of a certain drug in a patient's bloodstream t minutes after injection is given by SOt C(I) = 1 2 + 25 (a) Find the horizontal asymptote of C(I). What happens to the concentration of the drug as t increases? fil (b) Using our graphing utility, graph C(t) . .. (c) Determme the time at which the concentration IS highest

Minimum Cost A rectangular area adjacent to a river is to be fenced in; no fence is needed on the river side. The enclosed area is to be 1000 square feet. Fencing for the side parallel to the river is $5 per linear foot, and fencing for the other two sides is $8 per linear foot; the four corner posts are $25 apiece. Let x be the length of one of the sides perpendicular to the river. (a) Write a function C(x) that describes the cost of the project. (b) What is the domain of C? 6tl (c) Use a graphing utility to graph C(x). (d) Find the dimensions of the cheapest enclosure. Source: www.uncwil.edulcourseslmalhlllhbIPandRirationa/1 rational.hlml

Doppler Effect The Doppler effect (named after Christian Doppler) is the change in the pitch (frequency) of the sound from a source (s) as heard by an observer (0) when one or both are in motion. If we assume both the source and the observer are moving in the same direction, the relationship is where f' = faC = ::) f' = perceived pitch by the observer fa = actual pitch of the source v = speed of sound in air (assume 772.4 mph) va = speed of the observer Vs = speed of the source Suppose you are traveling down the road at 45 mph and you hear an ambulance (with siren) coming toward you from the rear. The actual pitch of the siren is 600 hertz (Hz). (a) Write a function f'(v\) that describes this scenario. (b) If f' = 620 Hz, find the speed of the ambulance. 1 (c) Use a graphing utility to graph the function. (d) Verify your answer from part (b). Source: www.kettering.edul-drusselll

Minimizing Surface Area United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. See the illustration. lLYx , x (a) Express the surface area S of the box as a function of x. (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area?

Minimizing Surface Area United Parcel Service has contracted you to design an open box with a square base that has a volume of 5000 cubic inches. See the illustration. lLYx ' x (a) Express the surface area S of the box as a function of x. lMl (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area?

Cost of a Can A can in the shape of a right circular cylinder is required to have a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6 per square centimeter, while the sides are made of material that costs 4 per square centimeter. (a) Express the total cost C of the material as a function of the radius r of the cylinder. (Refer to Figure 38.) ? (b) Graph C = C( r). For what value of r is the cost C a " minimum?

Material Needed to Make a Drum A steel drum in the shape of a right circular cylinder is required to have a volume of 100 cubic feet. (a) Express the amount A of material required to make the drum as a function of the radius r of the cylinder. (b) How much material is required if the drum's radius is 3 feet? (c) How much material is required if the drum's radius is 4 feet? (d) How much material is required if the drum's radius is 5 feet? (e) Graph A = A(r). For what value of r is A smallest?

Graph each of the following functions: x2-1 x3-1 y= x-I y= x - I X4 - 1 y=-x-I x 5 - 1 y= --x-I Is x = 1 a vertical asymptote? Why not? What is happening . x" - 1 for x = I? What do you conjecture about y = ---, x-I n 2: 1 an integer, for x = I?

Graph each of the following functions: x 2 y= -x-I X 4 y= -x-I x 6 y=-x-I xS y= -x-I What similarities do you see? What differences?

Write a few paragraphs that provide a general strategy for graphing a rational function. Be sure to mention the following: proper, improper, intercepts, and asymptotes.

Create a rational function that has the following characteristics: crosses the x-axis at 2; touches the x-axis at -1; one vertical asymptote at x = -5 and another at x = 6; and one horizontal asymptote, y = 3. Compare your function to a fellow classmate's. How do they differ? What are their similarities?

Create a rational function that has the following characteristics: crosses the x-axis at 3; touches the x-axis at - 2; one vertical asymptote, x = 1; and one horizontal asymptote, y = 2. Give your rational function to a fellow classmate and ask for a written critique of your rational function.

Create a rational function with the following characteristics: three real zeros, one of multiplicity 2;y-intercept 1; vertical asymptotes x = -2 and x = 3; oblique asymptote y = 2x + l. Is this rational function unique? Compare your function with those of other students. What will be the same as everyone else's? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the rational function?

Solve the inequality: 3 - 4x > 5. Graph the solution set. (pp. 124-131)

True or False The first step in solving the inequality x2 + 4x 2:: -4 is to factor the expression x2 + 4x.

In Problems 3-40, solve each inequality. (x - 5)2(x + 2) < 0

In Problems 3-40, solve each inequality. (x - 5)(x + 2f > 0

In Problems 3-40, solve each inequality. x3 - 4x2 > 0

In Problems 3-40, solve each inequality. x3 + 8x2 < 0

In Problems 3-40, solve each inequality. x3 - 9x :::; 0

In Problems 3-40, solve each inequality. x3 - x 2:: 0

In Problems 3-40, solve each inequality. 2x3 > -8x2

In Problems 3-40, solve each inequality. 3x3 < -15x2

In Problems 3-40, solve each inequality. (x - 1 )(x2 + X + 4) 2:: 0

In Problems 3-40, solve each inequality. (x + 2) (x2 - X + 1) 2:: 0

In Problems 3-40, solve each inequality. (x - l)(x - 2)(x - 3) :::; 0

In Problems 3-40, solve each inequality. (x + l)(x + 2)(x + 3) :::; 0

In Problems 3-40, solve each inequality. x3 - 2x2 - 3x > 0

In Problems 3-40, solve each inequality. x3 + 2x2 - 3x > 0

In Problems 3-40, solve each inequality. x4 > x2

In Problems 3-40, solve each inequality. X4 < 9x2

In Problems 3-40, solve each inequality. X4 > 1

In Problems 3-40, solve each inequality. x3 > 1

In Problems 3-40, solve each inequality. x + 1 --> 0 x - I

In Problems 3-40, solve each inequality. X - 3 -->0 x + 1

In Problems 3-40, solve each inequality. x - 1)(x + 1) :::; 0 x

In Problems 3-40, solve each inequality. (x - 3)(x + 2) :::; 0 x - I

In Problems 3-40, solve each inequality. (x - 2)2 2 2:: 0 x - I

In Problems 3-40, solve each inequality. (x + 5)2 2:: 0 x2 - 4

In Problems 3-40, solve each inequality. 6 6 x - ) < x

In Problems 3-40, solve each inequality. x+ -<7 x

In Problems 3-40, solve each inequality. x + 4 1 x-2

In Problems 3-40, solve each inequality. x + 2 1 x-4

In Problems 3-40, solve each inequality. 3x - 5 2 x+2

In Problems 3-40, solve each inequality. x - 4 --- 1 2x + 4

In Problems 3-40, solve each inequality. 1 2 --x - 2 <--3x - 9

In Problems 3-40, solve each inequality. 5 3 --x-3 >--x+1

In Problems 3-40, solve each inequality. 2x +5 x+1 --x+1 >--x-I

In Problems 3-40, solve each inequality. 1 3 --x+2 >--x+1

In Problems 3-40, solve each inequality. x2(3 + x)(x + 4) 0 (x + 5)(x - 1)

In Problems 3-40, solve each inequality. (x2 + l)(x - 2) 0 (x - l)(x + 1)

In Problems 3-40, solve each inequality. (3 - x)3(2x + 1) 3 < 0 x - I

In Problems 3-40, solve each inequality. (2 - x)3(3x - 2) -----:3:----- < 0 X + 1

For what positive numbers will the cube of a number exceed four times its square?

For what positive numbers will the cube of a number be less than the number?

What is the domain of the function f(x) = ?

What is the domain of the function f(x) = V - 3?

What is the domain of the function f(x) = ) x - 2? x + 4

What is the domain of the function f(x) = ) x - 1 ? x+4

In Problems 47-50, determine where the graph of f is below the graph of g by solving the inequality f(x) g(x). Graph f and g togethel: f(x) = X4 - 1 g(x) = _2x2 + 2

In Problems 47-50, determine where the graph of f is below the graph of g by solving the inequality f(x) g(x). Graph f and g togethel: f ( x) = X4 - 1 g(x) = x-I

In Problems 47-50, determine where the graph of f is below the graph of g by solving the inequality f(x) g(x). Graph f and g togethel: f(x) = X4 - 4 g(x) = 3x2

In Problems 47-50, determine where the graph of f is below the graph of g by solving the inequality f(x) g(x). Graph f and g togethel: f(x) = X4 g(x) = 2 - x2

Average Cost Suppose that the daily cost C of manufacturing bicycles is given by C(x) = 80x + 5000. Then the . - .. - 80x + 5000 average dally cost C IS given by C(x) = . How x many bicycles must be produced each day for the average cost to be no more than $100?

Average Cost See Problem 51. Suppose that the government imposes a $1000 per day tax on the bicycle manufacturer so that the daily cost C of manufacturing x bicycles is now given by C(x) = 80x + 6000. Now the average daily - - 80x + 6000 cost C is given by C(x) = . How many bicycles x must be produced each day for the average cost to be no more than $100?

Bungee "umping Originating on Pentecost Island in the Pacific, the practice of a person jumping from a high place harnessed to a flexible attachment was introduced to Western culture in 1979 by the Oxford University Dangerous Sport Club. One important parameter to know before attempting a bungee jump is the amount the cord will stretch at the bottom of the fall. The stiffness of the cord is related to the amount of stretch by the equation where 2W(S + L) K = -'---:: ,--- "":" S2 W = weight of the jumper (pounds) K = cord's stiffness (pounds per foot) L = free length of the cord (feet) S = stretch (feet) (a) A ISO-pound person plans to jump off a ledge attached to a cord of length 42 feet. If the stiffness of the cord is no less than 16 pounds per foot, how much will the cord stretch? (b) If safety requirements will not permit the jumper to get any closer than 3 feet to the ground, what is the minimum height required for the ledge in part (a)? Source: American Institute of Physics, Physics News Update, No. 150, November 5, 1993

Gravitational Force According to Newton's Law of universal gravitation, the attractive force F between two bodies is given by where F = G m1m2 , .2 m1, m2 are the masses of the two bodies r = distance between the Two bodies G = gravitational constant 6.6742 X 10-1 1 newtons meter2 kilogram-2 Suppose an object is traveling directly from Earth to the moon. The mass of the Earth is 5.9742 X 1024 kilograms, the mass of the moon is 7.349 X 1022 kilograms and the mean distance from Earth to the moon is 384,400 kilometers. For an object between Earth and the moon, how far from Earth is the force on the object due to the moon greater than the force on the object due to Earth? Source: www.solarviews.com;en.wikipedia.org

Make up an inequality that has no solution. Make up one that has exactly one solution.

The inequality X4 + 1 < -5 has no solution. Explain why.

A student attempted to solve the Inequality -- 0 by x - 3 multiplying both sides of the inequality by x - 3 to get x + 4 O. This led to a solution of {xix -4J. Is the student correct? Explain.

Write a rational inequality whose solution set is {xl -3 < x 5J.

Find f( -1) f(x) = 2x2 - x. (pp. 210-214)

Factor the expression 6x2 + x - 2. (pp. 49-55)

Find the quotient and remainder if 3x4 - 52 + 7x - 4 is divided by x - 3. (pp. 44-47 or 57-60)

Solve the equation x2 + x - 3 = O. (pp. 1 0 2-1 04)

In the process of polynomial division, ( Divisor) (Quotient) + ______ = ______.

When a polynomial function f is divided by x - c, the remainder is _____.

If a function f, whose domain is all real numbers, is even and if 4 is a zero of f, then _____ is also a zero.

True or False Every polynomial function of degree 3 with real coefficients has exactly three real zeros.

True or False The only potential rational zeros of f(x) = 2x5 - x3 + x2 - X + 1 are 1, 2.

True or False If f is a polynomial function of degree 4 and if f(2) = 5, then f(x) 5 x - 2 = p(x) + x - 2 where p(x) is a polynomial of degree 3.

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 4x3 - 3x2 - 8x + 4; x - 2

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = -4x3 + 5x2 + 8; x + 3

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 3x4 - 6x3 - 5x + 10; x - 2

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 4X4 - 15x2 - 4; x - 2

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 3x6 + 82x3 + 27; x + 3

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 2x6 - 18x4 + x2 - 9; x + 3

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 4x6 - 64x4 + x2 - 15; x + 4

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f (x) = x6 - 16x4 + x2 - 16; x + 4

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 2x4 - x3 + 2x - 1; x - 2"

In Problems 11-20, use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x - c is a factor of f(x). f(x) = 3x4 + x3 - 3x + 1; x +

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = -4x7 + x3 - x2 + 2

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = 5x4 + 2x2 - 6x - 5

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = 2x6 - 3x2 - X + 1

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = -3x5 + 4X4 + 2

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f (x) = 3x3 - 2x2 + X + 2

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f ( x) = -x3 - x2 + X + 1

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f ( x) = -x4 + x2 - 1

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = x4 + 5x3 - 2

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = x5 + X4 + x2 + X + 1

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = x5 - x4 + x3 - x2 + x - 1

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f (x) = x6 - 1

In Problems 21-32, tell the maximum number of real zeros that each polynomial function may have. Then use Descartes ' Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. f ( x) = x6 + 1

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = 3x4 - 3x3 + x2 - X + 1

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = x5 - X4 + 2X2 + 3

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = x5 - 6x2 + 9x - 3

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f ( x) = 2x5 - X4 - x2 + 1

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = -4x3 - x2 + X + 2

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = 6x4 - x2 + 2

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = 6x4 - x2 + 9

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = -4x3 + x2 + X + 6

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = 2x5 - x3 + 2x2 + 12

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = 3x5 - x2 + 2x + 18

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = 6x4 + 2x3 - x2 + 20

In Problems 33-44, list the potential rational zeros of each polynomial function. Do not attempt to find the zeros. f(x) = -6x3 - x2 + X + 10

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = x3 + 2x2 - 5x - 6

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = x3 + 8x2 + llx - 20

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = 2x3 - x2 + 2x - 1

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = 2x3 + x2 + 2x + 1

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = 2x3 - 4x2 - lOx + 20

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = 3x3 + 6x2 - 15x - 30

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f ( x) = 2X4 + x3 - 7 x2 - 3x + 3

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f ( x) = 2X4 - x3 - 5x2 + 2x + 2

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f ( x) = X4 + x3 - 3x2 - X + 2

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = x4 - x3 - 6x2 + 4x + 8

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = 4X4 + 5x3 + 9x2 + lOx + 2

In Problems 45-56, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. f(x) = 3x4 + 4x3 + 7x2 + 8x + 2

In Problems 57-68, solve each equation in the real number system . x4 - x3 + 2x2 - 4x - 8 = 0

In Problems 57-68, solve each equation in the real number system . 2x3 + 3x2 + 2x + 3 = 0

In Problems 57-68, solve each equation in the real number system . 3x3 + 4x2 - 7x + 2 = 0

In Problems 57-68, solve each equation in the real number system . 2x3 - 3x2 - 3x - 5 = 0

In Problems 57-68, solve each equation in the real number system . 3x3 - x2 - 15x + 5 = 0

In Problems 57-68, solve each equation in the real number system . 2x3 - l lx2 + lOx + 8 = 0

In Problems 57-68, solve each equation in the real number system . X4 + 4x3 + 2x2 - X + 6 = 0

In Problems 57-68, solve each equation in the real number system . X4 - 2x3 + lOx2 - 18x + 9 = 0

In Problems 57-68, solve each equation in the real number system . O - "3 x + "3 x + 1 = 0

In Problems 57-68, solve each equation in the real number system . x3 + -x2 + 3x - 2 = 0

In Problems 57-68, solve each equation in the real number system . 2x4 - 19x3 + 57x2 - 64x + 20 = 0

In Problems 57-68, solve each equation in the real number system . 2x4 + x3 - 24x2 + 20x + 16 = 0

In Problems 69-80, graph each polynomial function. f(x) = x3 + 2x2 - 5x - 6

In Problems 69-80, graph each polynomial function. f(x) = x3 + 8x2 + 1 1x - 20

In Problems 69-80, graph each polynomial function. f(x) = 2x3 - x2 + 2x - 1

In Problems 69-80, graph each polynomial function. f(x) = 2x3 + x2 + 2x + 1

In Problems 69-80, graph each polynomial function. f ( x) = X4 + x2 - 2

In Problems 69-80, graph each polynomial function. f(x) = X4 - 3x2 - 4

In Problems 69-80, graph each polynomial function. f(x) = 4X4 + 7x2 - 2

In Problems 69-80, graph each polynomial function. f(x) = 4X4 + 15x2 - 4

In Problems 69-80, graph each polynomial function. f(x) = X4 + x3 - 3x2 - X + 2

In Problems 69-80, graph each polynomial function. f(x) = X4 - x3 - 6x2 + 4x + 8

In Problems 69-80, graph each polynomial function. f(x) = 4x5 - 8x4 - X + 2

In Problems 69-80, graph each polynomial function. f(x) = 4x5 + 12x4 - X - 3

In Problems 81-88, find bounds on the real zeros of each polynomial function. f(x) = X4 - 3x2 - 4

In Problems 81-88, find bounds on the real zeros of each polynomial function. f(x) = X4 - 5x2 - 36

In Problems 81-88, find bounds on the real zeros of each polynomial function. f ( x) = X4 + x3 - x - I

In Problems 81-88, find bounds on the real zeros of each polynomial function. f ( x) = X4 - x3 + x - I

In Problems 81-88, find bounds on the real zeros of each polynomial function. f(x) = 3x4 + 3x3 - x2 - 12x - 12

In Problems 81-88, find bounds on the real zeros of each polynomial function. f(x) = 3x4 - 3x3 - 5x2 + 27x - 36

In Problems 81-88, find bounds on the real zeros of each polynomial function. f(x) = 4x5 - X4 + 2x3 - 2X2 + x - I

In Problems 81-88, find bounds on the real zeros of each polynomial function. f(x) = 4x5 + x4 + x3 + x2 - 2x - 2

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. f(x) = 8x4 - 2x2 + 5x - 1; [O, lJ

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. f(x) = x4 + 8x3 - x2 + 2; [ -1, OJ

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. f(x) = 2x3 + 6x2 - 8x + 2; [ -5, -4J

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. f(x) = 3x3 - LOx + 9; [ -3, -2J

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. f(x) = x5 - X4 + 7x3 - 7x2 - 18x + 18; [ 1.4, 1.5J

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. f(x) = xS - 3x4 - 2x3 + 6x2 + X + 2; [1.7, 1.8J

In Problems 95-98, each equation has a solution r in the interval indicated. Use the method of Example 10 to approximate this solution correct 10 two decimal places. 8x4 - 2x2 + 5x - 1 = 0; O:s r :s 1

In Problems 95-98, each equation has a solution r in the interval indicated. Use the method of Example 10 to approximate this solution correct 10 two decimal places. x4 + 8x3 - x2 + 2 = 0; - 1 :s r :s 0

In Problems 95-98, each equation has a solution r in the interval indicated. Use the method of Example 10 to approximate this solution correct 10 two decimal places. 2x3 + 6x2 - 8x + 2 = 0; -5 :s r :s -4

In Problems 95-98, each equation has a solution r in the interval indicated. Use the method of Example 10 to approximate this solution correct 10 two decimal places. 3x3 - LOx + 9 = 0; -3 :s r :s -2

In Problems 99-102, each polynomial function has exactly one positive zero. Use the method of Example 10 to approximate the zero correct to two decimal places. f(x) = x3 + x2 + X - 4

In Problems 99-102, each polynomial function has exactly one positive zero. Use the method of Example 10 to approximate the zero correct to two decimal places. f(x) = 2X4 + x2 - 1

In Problems 99-102, each polynomial function has exactly one positive zero. Use the method of Example 10 to approximate the zero correct to two decimal places. f(x) = 2X4 - 3x3 - 4x2 - 8

In Problems 99-102, each polynomial function has exactly one positive zero. Use the method of Example 10 to approximate the zero correct to two decimal places. f(x) = 3x3 - 2x2 - 20

Find k such that f(x) = x3 - kx2 + kx + 2 factor x - 2.

Find Ie such that f(x) = X4 - kx3 + kx2 + 1 factor x + 2.

What is the remainder when f(x) = 2x20 - 8xl0 + X - 2 is divided by x - I?

What is the remainder when f(x) = -3X 1 7 + x9 - XS + 2x is divided by x + I?

Use the Factor Theorem to prove that x - c is a factor of x" - e " for any positive integer n.

Use the Factor Theorem to prove that x + c is a factor of x" + e " if n 1 is an odd integer.

One solution of the equation x3 - 8x2 + 16x - 3 = 0 is 3. Find the sum of the remaining solutions

One solution of the equation x3 + 5x2 + 5x - 2 = 0 is -2. Find the sum of the remaining solutions.

Geometry What is the length of the edge of a cube if, after a slice 1 inch thick is cut from one side, the volume remaining is 294 cubic inches?

Geometry What is the length of the edge of a cube if its volume could be doubled by an increase of 6 centimeters in one edge, an increase of 12 centimeters in a second edge, and a decrease of 4 centimeters in the third edge?

Let f(x) be a polynomial function whose coefficients are integers. Suppose that r is a real zero of f and that the leading coefficient of f is 1. Use the Rational Zeros Theorem to show that r is either an integer or an irrational number.

Prove the Rational Zeros Theorem. [Hint: Let E, where p and q have no common factors except q 1 and -1, be a zero of the polynomial function f(x) = aw'(l1 + al_lx"-1 + ... + ajX + ao whose coefficients are all integers. Show that a " p" + a,,_IP"-lq + . . . + a1 pq"- l + aoq" = 0 Now, because p is a factor of the first n terms of this equation,p must also be a factor of the term aoqll. Since p is not a factor of q (why?),p must be a factor of ao. Similarly, q must be a factor of awJ

Bisection Method for Approximating Zeros of a Function f We begin with two consecutive integers, a and a + 1, such that f(a) and f(a + 1) are of opposite sign. Evaluate f at the midpoint Yi1 1 of a and a + 1. If f(l11l ) = 0, then 111 1 is the zero of f, and we are finished. Otherwise,f (ml) is of opposite sign to either f(a) or f(a + 1). Suppose that it is f(a) and f(ml) that are of opposite sign. Now evaluate f at the midpoint /112 of a and 111 1 ' Repeat this process until the desired degree of accuracy is obtained. Note that each iteration places the zero in an interval whose length is half that of the previous interval. Use the bisection method to solve Problems 95-102. [Hint: The process ends when both endpoints agree to the desired number of decimal places.]

Is 3" a zero of f(x) = 2x' + 3x - 6x + 7? Explain.

IS 3 " a zero of f(x) = 4x' - 5x - 3x + 1? Explain.

Is a zero of f(x) = 2x6 - 5x4 + x3 - X + I? Explain.

Is a zero of f(x) = x 7 + 6x5 - X4 + X + 2? Explain.

Find the sum and the product of the complex numbers 3 - 2i and -3 + 5i. (pp. 109-1 1 4)

In the complex number system, solve x2 + 2x + 2 = O. (pp. 1 l 4--1 16)

Every polynomial function of odd degree with real coefficients will have at least ____ real zeroes).

If 3 + 4i is a zero of a polynomial function of degree 5 with real coefficients, then so is ____.

True or False A polynomial function of degree n with real coefficients has exactly n complex zeros. At most n of them are real zeros

True or False A polynomial function of degree 4 with real coefficients could have -3, 2 + i, 2 - i, and -3 + 5i as its zeros.

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 3; zeros: 3, 4 - i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 3; zeros: 4, 3 + i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 4; zeros: i, 1 + i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 4; zeros: 1, 2,2 + i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 5; zeros: 1, i, 2i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 5; zeros: 0, 1, 2, i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 4; zeros: i, 2, -2

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 4; zeros: 2 - i, -i

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 6; zeros: 2, 2 + i, -3 - i, 0

In Problems 7-16, information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 6; zeros: i, 3 - 2i, -2 + i

In Problems 17-22, form a polynomial f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree 4; zeros: 3 + 2i; 4, multiplicity 2

In Problems 17-22, form a polynomial f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree 4; zeros: i, 1 + 2i

In Problems 17-22, form a polynomial f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree 5; zeros: 2; -i; 1 + i

In Problems 17-22, form a polynomial f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree 6; zeros: i, 4 - i; 2 + i

In Problems 17-22, form a polynomial f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree 4; zeros: 3, multiplicity 2; -i

In Problems 17-22, form a polynomial f(x) with real coefficients having the given degree and zeros. Answers will vary depending on the choice of leading coefficient. Degree 5; zeros: 1, multiplicity 3; 1 + i

In Problems 23-30, use the given zero to find the remaining zeros of each function. f(x) = .2 - 4 + 4x - 16; zero: 2i

In Problems 23-30, use the given zero to find the remaining zeros of each function. g(x) = .2 + 3 + 25x + 75; zero: -5i

In Problems 23-30, use the given zero to find the remaining zeros of each function. f(x) = 2x4 + 5.2 + 5 + 20x - 12; zero: -2i

In Problems 23-30, use the given zero to find the remaining zeros of each function. h(x) = 3x4 + 5.2 + 25 + 45x - 18; zero: 3i

In Problems 23-30, use the given zero to find the remaining zeros of each function. h(x) = X4 - 9.2 + 21 + 21x - 130; zero: 3 - 2i

In Problems 23-30, use the given zero to find the remaining zeros of each function. f(x) = x4 - 7.2 + 14 - 38x - 60; zero: 1 + 3i

In Problems 23-30, use the given zero to find the remaining zeros of each function. h( x) = 3Xi + 2x4 + 15.2 + 10 - 528x - 352; zero: -4i

In Problems 23-30, use the given zero to find the remaining zeros of each function. g(x) = 2Xi - 3x4 - 5.2 - 15 - 207x + 108; zero: 3i

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = .2 - 1

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = X4 - 1

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = .2 - 8 + 25x - 26

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = .2 + 13 + 57x + 85

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = X4 + 5 + 4

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = x4 + 13 + 36

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = x4 + 2.2 + 22 + 50x - 75

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = x4 + 3.2 - 19 + 27x - 252

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = 3x4 - .2 - 9 + 159x - 52

In Problems 31-40, find the complex zeros of each polynomial function. Write f in factored form. f(x) = 2X4 + .2 - 35x2 - 1 13x + 65

In Problems 41 and 42, explain why the facts given are contradictory. f(x) is a polynomial of degree 3 whose coefficients are real numbers; its zeros are 4 + i, 4 - i, and 2 + i.

In Problems 41 and 42, explain why the facts given are contradictory. f(x) is a polynomial of degree 3 whose coefficients are real numbers; its zeros are 2, i, and 3 + i.

f(x) is a polynomial of degree 4 whose coefficients are real numbers; three of its zeros are 2, 1 + 2i, and 1 - 2i. Explain why the remaining zero must be a real number.

f(x) is a polynomial of degree 4 whose coefficients are real numbers; two of its zeros are -3 and 4 - i. Explain why one of the remaining zeros must be a real number. Write down one of the missing zeros.

In Problems 1-4, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = 4x5 - 3x2 + 5x - 2

In Problems 1-4, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = 2x+ 1

In Problems 1-4, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = 3x2 + 5xl/2 - 1

In Problems 1-4, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. f(x) = 3

In Problems 5-10, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. f(x) = (x + 2?

In Problems 5-10, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. f(x) = -x3 + 3

In Problems 5-10, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. f(x) = - (x - 1 )4

In Problems 5-10, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. f(x) = (x - 1 )4 - 2

In Problems 5-10, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. f(x) = (x - 1 )4 + 2

In Problems 5-10, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. f(x) = (1 - x)3

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x(x + 2)(x + 4)

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = x(x - 2)(x - 4)

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 2)2(x + 4)

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 2)(x + 4f

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = -2x3 + 4x2

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = -4x3 + 4x

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 1 f(x + 3) (x + 1)

In Problems 11-18: (a) Find the x- and y-intercepts of each polynomial function f. (b) Determine whether the graph of f touches or crosses the x-axis at each x-intercept. (c) End behavior:find the power function that the graph off resembles for large values of Ix!. (d) Determine the maximum number of turning points of the graph of .f. (e) Determine the behavior of the graph off near each x-intercept. (f) Put all the information together to obtain the graph off (You may need to locate additional points on the graph.) f(x) = (x - 4)(x + 2)2(x - 2)

In Problems 19-22, find the domain of each rational function. Find any horizontal, vertical, or oblique asymptotes. R(x) = x + 2

In Problems 19-22, find the domain of each rational function. Find any horizontal, vertical, or oblique asymptotes. R(x) = x2 + 4

In Problems 19-22, find the domain of each rational function. Find any horizontal, vertical, or oblique asymptotes. R(x) = x2 + 3x + 2

In Problems 19-22, find the domain of each rational function. Find any horizontal, vertical, or oblique asymptotes. R(x) = x3

In Problems 23-34, discuss each rational function following the seven steps given on page 355. R(x) = --x 2x - 6

In Problems 23-34, discuss each rational function following the seven steps given on page 355. R(x) =--x

In Problems 23-34, discuss each rational function following the seven steps given on page 355. H(x) = x x( ) -2

In Problems 23-34, discuss each rational function following the seven steps given on page 355. H ( x) = x2 _ 1

In Problems 23-34, discuss each rational function following the seven steps given on page 355. R(x) = x + x - 6 x - x - 6

In Problems 23-34, discuss each rational function following the seven steps given on page 355. R(x) = x2 - 6: + 9

In Problems 23-34, discuss each rational function following the seven steps given on page 355. F(x) = - 2-- X - 4

In Problems 23-34, discuss each rational function following the seven steps given on page 355. F(x) = (x - 1)

In Problems 23-34, discuss each rational function following the seven steps given on page 355. R(x) = -:: (x - 1 )2

In Problems 23-34, discuss each rational function following the seven steps given on page 355. R(x) = x - 9

In Problems 23-34, discuss each rational function following the seven steps given on page 355. G (x) = -2::---- X - X - 2

In Problems 23-34, discuss each rational function following the seven steps given on page 355. F (x) = -'-- x2 - 1

In Problems 35-44, solve each inequality. Graph the solution set. x3 + x2 < 4x + 4

In Problems 35-44, solve each inequality. Graph the solution set. x3 + 4x2 2: X + 4

In Problems 35-44, solve each inequality. Graph the solution set. 6 --2: 1 x + 3

In Problems 35-44, solve each inequality. Graph the solution set. -2 1 _ 3x < 1

In Problems 35-44, solve each inequality. Graph the solution set. 2x - 6 < 2 1 - X

In Problems 35-44, solve each inequality. Graph the solution set. 3 - 2x .--- 2:2 2x + 5

In Problems 35-44, solve each inequality. Graph the solution set. (x - 2)(x - 1) 2: 0 x - 3

In Problems 35-44, solve each inequality. Graph the solution set. x + 1 . ::; 0 x(x - 5

In Problems 35-44, solve each inequality. Graph the solution set. x2 - 8x + 12 x2 _ 16 > 0

In Problems 35-44, solve each inequality. Graph the solution set. x(x2 + X - 2) ::; 0 x2 + 9x + 20

In Problems 45-48, find the remainder R when f(x) is divided by g(x). Is g a faclOr of f? f(x) = 8x3 - 3x2 + X + 4; g(x) = x - I

In Problems 45-48, find the remainder R when f(x) is divided by g(x). Is g a faclOr of f? f(x) = 2x3 + 8x2 - 5x + 5; g(x) = x - 2

In Problems 45-48, find the remainder R when f(x) is divided by g(x). Is g a faclOr of f? f(x) = X4 - 2x3 + 15x - 2; g(x) = x + 2

In Problems 45-48, find the remainder R when f(x) is divided by g(x). Is g a faclOr of f? f(x) = X4 - x2 + 2x + 2; g(x) = x + 1

Find the value of f(x) = 12x6 - 8x4 + 1 at x = 4.

Find the value of f(x) = - 16x3 + 1 8x 2 - X + 2 at x = -2.

In Problems 51 and 52, use Descartes ' Rule of Signs to determine how many positive and negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = 12x8 - x7 + 8x4 - 2x3 + X + 3

In Problems 51 and 52, use Descartes ' Rule of Signs to determine how many positive and negative zeros each polynomial function may have. Do not attempt to find the zeros. f(x) = -6x5 + X4 + 5x3 + X + 1

List all the potential rational zeros of f(x) = 12x8 - x7 + 6x4 - x3 + X - 3.

List all the potential rational zeros of f(x) = -6x5 + X 4 + 2x3 - X + 1.

In Problems 55-60, use the Rational Zeros Theorem 10 find all the real zeros of each polynomial function. Use the zeros 10 faclOr f over the real numbers f(x) = x3 - 3x2 - 6x + 8

In Problems 55-60, use the Rational Zeros Theorem 10 find all the real zeros of each polynomial function. Use the zeros 10 faclOr f over the real numbers. f(x) = x3 - x2 - lOx - 8

In Problems 55-60, use the Rational Zeros Theorem 10 find all the real zeros of each polynomial function. Use the zeros 10 faclOr f over the real numbers. f(x) = 4x3 + 4x2 - 7x + 2

In Problems 55-60, use the Rational Zeros Theorem 10 find all the real zeros of each polynomial function. Use the zeros 10 faclOr f over the real numbers. f(x) = 4x3 - 4x2 - 7x - 2

In Problems 55-60, use the Rational Zeros Theorem 10 find all the real zeros of each polynomial function. Use the zeros 10 faclOr f over the real numbers. f(x) = X4 - 4x3 + 9x2 - 20x + 20

In Problems 55-60, use the Rational Zeros Theorem 10 find all the real zeros of each polynomial function. Use the zeros 10 faclOr f over the real numbers. f(x) = X4 + 6x3 + l lx2 + 12x + 18

In Problems 61--64, solve each equation in the real number system. 2X4 + 2x3 - llx2 + X - 6 = 0

In Problems 61--64, solve each equation in the real number system. 3x4 + 3x3 - 1 7x2 + X - 6 = 0

In Problems 61--64, solve each equation in the real number system. 2X4 + 7 x3 + x2 - 7 x - 3 = 0

In Problems 61--64, solve each equation in the real number system. 2X4 + 7x3 - 5x2 - 28x - 12 = 0

In Problems 65-68, find bounds 10 the real zeros of each polynomial function. f(x) = x3 - x2 - 4x + 2

In Problems 65-68, find bounds 10 the real zeros of each polynomial function. f(x) = x3 + x 2 - lOx - 5

In Problems 65-68, find bounds 10 the real zeros of each polynomial function. f(x) = 2x3 - 7x2 - lOx + 35

In Problems 65-68, find bounds 10 the real zeros of each polynomial function. f(x) = 3x3 - 7x2 - 6x + 14

In Problems 69-72, use the Intermediate Value Theorem to show that each polynomial has a zero in the given interval. f(x) = 3x3 - x-I; [0, 1]

In Problems 69-72, use the Intermediate Value Theorem to show that each polynomial has a zero in the given interval. f(x) = 2x3 - x 2 - 3; [1, 2]

In Problems 69-72, use the Intermediate Value Theorem to show that each polynomial has a zero in the given interval. f(x) = 8x4 - 4x3 - 2x - 1; [0, 1]

In Problems 69-72, use the Intermediate Value Theorem to show that each polynomial has a zero in the given interval. f(x) = 3x4 + 4x3 - 8x - 2; [1, 2]

In Problems 73-76, each polynomial has exactly one positive zero. Approximate the zero correct to two decimal places. f(x) = x3 - X - 2

In Problems 73-76, each polynomial has exactly one positive zero. Approximate the zero correct to two decimal places. f(x) = 2x3 - x 2 - 3

In Problems 73-76, each polynomial has exactly one positive zero. Approximate the zero correct to two decimal places. f(x) = 8x4 - 4x3 - 2x - 1

In Problems 73-76, each polynomial has exactly one positive zero. Approximate the zero correct to two decimal places. f(x) = 3x4 + 4x3 - 8x - 2

In Problems 77-80, information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial with real coefficients that has the zeros. Degree 3; zeros: 4 + i, 6

In Problems 77-80, information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial with real coefficients that has the zeros. Degree 3; zeros: 3 + 4i, 5

In Problems 77-80, information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial with real coefficients that has the zeros. Degree 4; zeros: i, 1 + i

In Problems 77-80, information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial with real coefficients that has the zeros. Degree 4; zeros: 1, 2, 1 + i

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form.

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = x3 - x2 - lOx - 8

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = 4x3 + 4x2 - 7x + 2

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = 4x3 - 4x2 - 7x - 2

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = X4 - 4x3 + 9x2 - 20x + 20

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = X4 + 6x3 + llx2 + 12x + 18

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = 2X4 + 2x3 - llx2 + X - 6

In Problems 81-88, find the complex zeros of each polynomial function f(x). Write f in factored form. f(x) = 3x4 + 3x3 - 17x2 + X - 6

Making a Can A can in the shape of a right circular cylinder is required to have a volume of 250 cubic centimeters. (a) Express the amount A of material to make the can as a function of the radius r of the cylinder. (b) How much material is required if the can is of radius 3 centimeters?(c) How much material is required if the can is of radius 5 centimeters? (d) Graph A = A(r). For what value of r is A smallest?