Solution: Find all rational, irrational, and complex zeros

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function.f1x2 x 2  12x  12 2  4x  1

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function.f1x2 2x f1x2 x 2  12x  12 2

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function.g1x2 1  8x x 2

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function.g1x2 6x 3x 2 g1

Find the maximum or minimum value of the quadratic function.f1x2 2x 2  4x 5

Find the maximum or minimum value of the quadratic function.g1x2 1 x x 2 f1

A stone is thrown upward from the top of a building. Its height (in feet) above the ground after t seconds is given by the function . What maximum height does the stone reach?

The profit P (in dollars) generated by selling x units of a certain commodity is given by the function What is the maximum profit, and how many units must be sold to generate it?P1x2 1500  12x 0.004x 2 1

Graph the polynomial by transforming an appropriate graph of the form y x n . Show clearly all x- and y-intercepts.P1x2 x 3 16 3  64 P

Graph the polynomial by transforming an appropriate graph of the form y x n . Show clearly all x- and y-intercepts.

Graph the polynomial by transforming an appropriate graph of the form y x n . Show clearly all x- and y-intercepts.P1x2 21x  12 4 32

Graph the polynomial by transforming an appropriate graph of the form y x n . Show clearly all x- and y-intercepts.P1x2 81 1x 32 4 P1

Graph the polynomial by transforming an appropriate graph of the form y x n . Show clearly all x- and y-intercepts.P1x2 32  1x 12 5  96 5

Graph the polynomial by transforming an appropriate graph of the form y x n . Show clearly all x- and y-intercepts.P1x2 31x  22 P1x2 32  1x 12 5  96 5

A polynomial function P is given. (a) Determine the multiplicity of each zero of P. (b) Sketch a graph of P.P1x2 x 3 1x 22 2 C

A polynomial function P is given. (a) Determine the multiplicity of each zero of P. (b) Sketch a graph of P.P1x2 x1x  12 3 1x 12 2 P

Use a graphing device to graph the polynomial. Find the x- and y-intercepts and the coordinates of all local extrema, correct to the nearest decimal. Describe the end behavior of the polynomial.P1x2 x 2 2 3 4x  1 P1

Use a graphing device to graph the polynomial. Find the x- and y-intercepts and the coordinates of all local extrema, correct to the nearest decimal. Describe the end behavior of the polynomial.P1x2 2x3  6x P1x2 x 2 2 3

Use a graphing device to graph the polynomial. Find the x- and y-intercepts and the coordinates of all local extrema, correct to the nearest decimal. Describe the end behavior of the polynomial.P1x2 3x 4 4x 3 10x 1 P1x

Use a graphing device to graph the polynomial. Find the x- and y-intercepts and the coordinates of all local extrema, correct to the nearest decimal. Describe the end behavior of the polynomial.P1x2 x 5  x 4 7x 3 x 2  6x  3 P

The strength S of a wooden beam of width x and depth y is given by the formula S 13.8xy 2 . A beam is to be cut from a log of diameter 10 in., as shown in the figure. (a) Express the strength S of this beam as a function of x only. (b) What is the domain of the function S? (c) Draw a graph of S. (d) What width will make the beam the strongest?

A small shelter for delicate plants is to be constructed of thin plastic material. It will have square ends and a rectangular top and back, with an open bottom and front, as shown in the figure. The total area of the four plastic sides is to be 1200 in2 . (a) Express the volume V of the shelter as a function of the depth x. (b) Draw a graph of V. (c) What dimensions will maximize the volume of the shelter?

Find the quotient and remainder.x 2 3x  5 x 2

Find the quotient and remainder.x 2  x 12 x 3

Find the quotient and remainder.x 3 x 2  11x  2 x 4

Find the quotient and remainder.x 3  2x 2 10 x  3

Find the quotient and remainder.x 4 8x 2  2x  7 x  5

Find the quotient and remainder.2x 4  3x 3 12 x  4

Find the quotient and remainder.2x 3  x 2 8x  15 x 2  2x 1 2

Find the quotient and remainder.x 4 2x 2  7x x 2 x  3

Find the indicated value of the polynomial using the Remainder Theorem.P1x2 2x P152 3 9x2 7x  13 1

Find the indicated value of the polynomial using the Remainder Theorem.Q1x2 x Q132 4  4x 3  7x 2  10x  15

Show that is a zero of the polynomial.P1x2 2x4  x3 5x2  10x 4 1

Use the Factor Theorem to show that x  4 is a factor of the polynomial.P1x2 x5  4x4 7x3 23x2  23x  12 P

What is the remainder when the polynomial is divided by x 1?P1x2 x500  6x201 x2 2x  4 P

What is the remainder when x101 x4  2 is divided by x  1?

A polynomial P is given. (a) List all possible rational zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes Rule of Signs.P1x2 x5 6x3 x2  2x  18 P

A polynomial P is given. (a) List all possible rational zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes Rule of Signs.P1x2 6x 4  3x 3  x 2  3x  4

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 x 3 16x

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 x 3 3x 2 4x P

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 x4  x3 2x2

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 x 4 5x 2  4

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 x4 2x3 7x2  8x  12 P

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 x4 2x3 2x2  8x 8 P1

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 2x4  x3  2x2 3x 2 P

A polynomial P is given. (a) Find all real zeros of P, and state their multiplicities. (b) Sketch the graph of P.P1x2 9x 5 21x 4  10x 3  6x 2 3x 1 P1

Evaluate the expression and write in the form a  bi.12 3i2  11  4i2

Evaluate the expression and write in the form a  bi.13 6i2 16 4i2

Evaluate the expression and write in the form a  bi.P12  i2 13 2i2 i

Evaluate the expression and write in the form a  bi.4i12 1 2 12  i2 13 2i2 i2

Evaluate the expression and write in the form a  bi.4  2i 2 i

Evaluate the expression and write in the form a  bi.8  3i 4  3i

Evaluate the expression and write in the form a  bi.i 25

Evaluate the expression and write in the form a  bi.11  i2 3

Evaluate the expression and write in the form a  bi.11 112 11  112 11

Evaluate the expression and write in the form a  bi.110 # 140

Find a polynomial of degree 3 with constant coefficient 12 and zeros , 2, and 3.

Find a polynomial of degree 4 that has integer coefficients and zeros 3i and 4, with 4 a double zero.

Does there exist a polynomial of degree 4 with integer coefficients that has zeros i, 2i, 3i, and 4i? If so, find it. If not, explain why

Prove that the equation 3x 4  5x 2  2 0 has no real root.

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 3 3x 2 13x  15 29

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 2x3  5x2 6x 9 P

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 4  6x 3  17x 2  28x  20

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 4  7x 3  9x 2 17x 20 P

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x5 3x4 x3  11x2 12x  4 P1

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 4 81

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 6 64

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 18x 3  3x 2 4x 1 P

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 6x 4 18x 3  6x 2 30x  36 P

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 4  15x 2  54

Use a graphing device to find all real solutions of the equation.. 2x 2 5x  3

Use a graphing device to find all real solutions of the equation.x 3  x 2 14x 24 0 7

Use a graphing device to find all real solutions of the equation.x 4 3x 3 3x 2 9x 2 0 74.

Use a graphing device to find all real solutions of the equation.x 5 x  3

A polynomial function P is given. Find all the real zeros of P, and factor P completely into linear and irreducible quadratic factors with real coefficientsP1x2 x 2 4 3 2x 4 61

A polynomial function P is given. Find all the real zeros of P, and factor P completely into linear and irreducible quadratic factors with real coefficients.P1x2 x4  3x P1x2 x 2 4 3

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.r1x2 3x 12 x  1 P

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.r1x2 1 1x  22 2

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.r1x2 x 2 x 2 2x 8 r1x

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.r1x2 2x 2 6x 7 x 4 r1x

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.r1x2 x 2 9 2x 2  1

Graph the rational function. Show clearly all x- and y-intercepts and asymptotes.r1x2 x 3  27 x  4

Use a graphing device to analyze the graph of the rational function. Find all x- and y-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.r1x2 x 3 2x  6

Use a graphing device to analyze the graph of the rational function. Find all x- and y-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.r1x2 2x 7 x2  9 r

Use a graphing device to analyze the graph of the rational function. Find all x- and y-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.r1x2 x 3  8 x 2 x 2 r

Use a graphing device to analyze the graph of the rational function. Find all x- and y-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.r1x2 2x3 x2 x  1 r

Find the coordinates of all points of intersection of the graphs of y x4  x2  24x and y 6x3  20