 3.3.1: Two forms of the Division Algorithm are shown below. Identify and l...
 3.3.2: In Exercises 25, fill in the blank(s). The rational expression p(x)...
 3.3.3: In Exercises 25, fill in the blank(s). Every rational zero of a pol...
 3.3.4: In Exercises 25, fill in the blank(s). The theorem that can be used...
 3.3.5: In Exercises 25, fill in the blank(s). A real number c is a(n) ____...
 3.3.6: How many negative real zeros are possible for a polynomial function...
 3.3.7: You divide the polynomial f(x) by (x 4) and obtain a remainder of 7...
 3.3.8: What value should you write in the circle to check whether (x 2) is...
 3.3.9: In Exercises 912, use long division to divide and use the result to...
 3.3.10: In Exercises 912, use long division to divide and use the result to...
 3.3.11: In Exercises 912, use long division to divide and use the result to...
 3.3.12: In Exercises 912, use long division to divide and use the result to...
 3.3.13: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.14: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.15: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.16: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.17: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.18: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.19: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.20: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.21: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.22: In Exercises 1322, use long division to divide. 13. (x3 4x2 17x + 6...
 3.3.23: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.24: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.25: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.26: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.27: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.28: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.29: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.30: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.31: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.32: In Exercises 2332, use synthetic division to divide. 23. (3x3 17x2 ...
 3.3.33: In Exercises 3336, use a graphing utility to graph the two equation...
 3.3.34: In Exercises 3336, use a graphing utility to graph the two equation...
 3.3.35: In Exercises 3336, use a graphing utility to graph the two equation...
 3.3.36: In Exercises 3336, use a graphing utility to graph the two equation...
 3.3.37: In Exercises 37 42, write the function in the form f(x) = (x k)q(x)...
 3.3.38: In Exercises 37 42, write the function in the form f(x) = (x k)q(x)...
 3.3.39: In Exercises 37 42, write the function in the form f(x) = (x k)q(x)...
 3.3.40: In Exercises 37 42, write the function in the form f(x) = (x k)q(x)...
 3.3.41: In Exercises 37 42, write the function in the form f(x) = (x k)q(x)...
 3.3.42: In Exercises 37 42, write the function in the form f(x) = (x k)q(x)...
 3.3.43: In Exercises 43 46, use the Remainder Theorem and synthetic divisio...
 3.3.44: In Exercises 43 46, use the Remainder Theorem and synthetic divisio...
 3.3.45: In Exercises 43 46, use the Remainder Theorem and synthetic divisio...
 3.3.46: In Exercises 43 46, use the Remainder Theorem and synthetic divisio...
 3.3.47: In Exercises 4752, use synthetic division to show that x is a solut...
 3.3.48: In Exercises 4752, use synthetic division to show that x is a solut...
 3.3.49: In Exercises 4752, use synthetic division to show that x is a solut...
 3.3.50: In Exercises 4752, use synthetic division to show that x is a solut...
 3.3.51: In Exercises 4752, use synthetic division to show that x is a solut...
 3.3.52: In Exercises 4752, use synthetic division to show that x is a solut...
 3.3.53: In Exercises 5358, (a) verify the given factor(s) of the function f...
 3.3.54: In Exercises 5358, (a) verify the given factor(s) of the function f...
 3.3.55: In Exercises 5358, (a) verify the given factor(s) of the function f...
 3.3.56: In Exercises 5358, (a) verify the given factor(s) of the function f...
 3.3.57: In Exercises 5358, (a) verify the given factor(s) of the function f...
 3.3.58: In Exercises 5358, (a) verify the given factor(s) of the function f...
 3.3.59: In Exercises 59 62, use the Rational Zero Test to list all possible...
 3.3.60: In Exercises 59 62, use the Rational Zero Test to list all possible...
 3.3.61: In Exercises 59 62, use the Rational Zero Test to list all possible...
 3.3.62: In Exercises 59 62, use the Rational Zero Test to list all possible...
 3.3.63: In Exercises 6366, use Descartess Rule of Signs to determine the po...
 3.3.64: In Exercises 6366, use Descartess Rule of Signs to determine the po...
 3.3.65: In Exercises 6366, use Descartess Rule of Signs to determine the po...
 3.3.66: In Exercises 6366, use Descartess Rule of Signs to determine the po...
 3.3.67: In Exercises 6772, (a) use Descartess Rule of Signs to determine th...
 3.3.68: In Exercises 6772, (a) use Descartess Rule of Signs to determine th...
 3.3.69: In Exercises 6772, (a) use Descartess Rule of Signs to determine th...
 3.3.70: In Exercises 6772, (a) use Descartess Rule of Signs to determine th...
 3.3.71: In Exercises 6772, (a) use Descartess Rule of Signs to determine th...
 3.3.72: In Exercises 6772, (a) use Descartess Rule of Signs to determine th...
 3.3.73: In Exercises 7376, use synthetic division to verify the upper and l...
 3.3.74: In Exercises 7376, use synthetic division to verify the upper and l...
 3.3.75: In Exercises 7376, use synthetic division to verify the upper and l...
 3.3.76: In Exercises 7376, use synthetic division to verify the upper and l...
 3.3.77: In Exercises 77 80, find the rational zeros of the polynomial funct...
 3.3.78: In Exercises 77 80, find the rational zeros of the polynomial funct...
 3.3.79: In Exercises 77 80, find the rational zeros of the polynomial funct...
 3.3.80: In Exercises 77 80, find the rational zeros of the polynomial funct...
 3.3.81: In Exercises 81 84, match the cubic function with the correct numbe...
 3.3.82: In Exercises 81 84, match the cubic function with the correct numbe...
 3.3.83: In Exercises 81 84, match the cubic function with the correct numbe...
 3.3.84: In Exercises 81 84, match the cubic function with the correct numbe...
 3.3.85: In Exercises 85 88, the graph of y = f(x) is shown. Use the graph a...
 3.3.86: In Exercises 85 88, the graph of y = f(x) is shown. Use the graph a...
 3.3.87: In Exercises 85 88, the graph of y = f(x) is shown. Use the graph a...
 3.3.88: In Exercises 85 88, the graph of y = f(x) is shown. Use the graph a...
 3.3.89: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.90: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.91: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.92: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.93: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.94: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.95: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.96: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.97: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.98: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.99: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.100: In Exercises 89 100, find all real zeros of the polynomial function...
 3.3.101: In Exercises 101 104, (a) use the zero or root feature of a graphin...
 3.3.102: In Exercises 101 104, (a) use the zero or root feature of a graphin...
 3.3.103: In Exercises 101 104, (a) use the zero or root feature of a graphin...
 3.3.104: In Exercises 101 104, (a) use the zero or root feature of a graphin...
 3.3.105: The table shows the numbers S of cellular phone subscriptions per 1...
 3.3.106: The numbers of employees E (in thousands) in education and health s...
 3.3.107: A rectangular package sent by a delivery service can have a maximum...
 3.3.108: The number of parts per million of nitric oxide emissions y from a ...
 3.3.109: In Exercises 109 and 110, determine whether the statement is true o...
 3.3.110: The value x = 1 7 is a zero of the polynomial function f(x) = 3x5 2...
 3.3.111: In Exercises 111 and 112, the graph of a cubic polynomial function ...
 3.3.112: In Exercises 111 and 112, the graph of a cubic polynomial function ...
 3.3.113: Let y = f(x) be a fourthdegree polynomial with leading coefficient...
 3.3.114: Find the value of k such that x 3 is a factor of x3 kx2 + 2kx 12.
 3.3.115: Complete each polynomial division. Write a brief description of the...
 3.3.116: A graph of y = f(x) is shown, where f(x) = 2x5 3x4 + x3 8x2 + 5x + ...
 3.3.117: In Exercises 117 120, use any convenient method to solve the quadra...
 3.3.118: In Exercises 117 120, use any convenient method to solve the quadra...
 3.3.119: In Exercises 117 120, use any convenient method to solve the quadra...
 3.3.120: In Exercises 117 120, use any convenient method to solve the quadra...
Solutions for Chapter 3.3: Polynomial and Rational Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 3.3: Polynomial and Rational Functions
Get Full SolutionsAlgebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. Chapter 3.3: Polynomial and Rational Functions includes 120 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 120 problems in chapter 3.3: Polynomial and Rational Functions have been answered, more than 58997 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Column space C (A) =
space of all combinations of the columns of A.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).