 3.1: In Exercises 1 6, sketch the graph of each function and compare it ...
 3.2: In Exercises 1 6, sketch the graph of each function and compare it ...
 3.3: In Exercises 1 6, sketch the graph of each function and compare it ...
 3.4: In Exercises 1 6, sketch the graph of each function and compare it ...
 3.5: In Exercises 1 6, sketch the graph of each function and compare it ...
 3.6: In Exercises 1 6, sketch the graph of each function and compare it ...
 3.7: In Exercises 710, describe the graph of the function and identify t...
 3.8: In Exercises 710, describe the graph of the function and identify t...
 3.9: In Exercises 710, describe the graph of the function and identify t...
 3.10: In Exercises 710, describe the graph of the function and identify t...
 3.11: In Exercises 11 and 12, write the standard form of the quadratic fu...
 3.12: In Exercises 11 and 12, write the standard form of the quadratic fu...
 3.13: A rectangle is inscribed in the region bounded by the xaxis, the y...
 3.14: A college has 1500 feet of portable rink boards to form three adjac...
 3.15: In Exercises 1518, sketch the graph of f(x) = x3 and the graph of t...
 3.16: In Exercises 1518, sketch the graph of f(x) = x3 and the graph of t...
 3.17: In Exercises 1518, sketch the graph of f(x) = x3 and the graph of t...
 3.18: In Exercises 1518, sketch the graph of f(x) = x3 and the graph of t...
 3.19: In Exercises 19 and 20, use a graphing utility to graph the functio...
 3.20: In Exercises 19 and 20, use a graphing utility to graph the functio...
 3.21: In Exercises 21 and 22, use the Leading Coefficient Test to describ...
 3.22: In Exercises 21 and 22, use the Leading Coefficient Test to describ...
 3.23: In Exercises 2328, (a) find the zeros algebraically, (b) use a grap...
 3.24: In Exercises 2328, (a) find the zeros algebraically, (b) use a grap...
 3.25: In Exercises 2328, (a) find the zeros algebraically, (b) use a grap...
 3.26: In Exercises 2328, (a) find the zeros algebraically, (b) use a grap...
 3.27: In Exercises 2328, (a) find the zeros algebraically, (b) use a grap...
 3.28: In Exercises 2328, (a) find the zeros algebraically, (b) use a grap...
 3.29: In Exercises 29 32, find a polynomial function that has the given z...
 3.30: In Exercises 29 32, find a polynomial function that has the given z...
 3.31: In Exercises 29 32, find a polynomial function that has the given z...
 3.32: In Exercises 29 32, find a polynomial function that has the given z...
 3.33: In Exercises 33 and 34, sketch the graph of the function by (a) app...
 3.34: In Exercises 33 and 34, sketch the graph of the function by (a) app...
 3.35: In Exercises 3538, (a) use the Intermediate Value Theorem and a gra...
 3.36: In Exercises 3538, (a) use the Intermediate Value Theorem and a gra...
 3.37: In Exercises 3538, (a) use the Intermediate Value Theorem and a gra...
 3.38: In Exercises 3538, (a) use the Intermediate Value Theorem and a gra...
 3.39: In Exercises 3944, use long division to divide. 39. 24x2 x 8 3x 2
 3.40: In Exercises 3944, use long division to divide. 40. 4x2 + 7 3x 2
 3.41: In Exercises 3944, use long division to divide. 41. x4 3x2 + 2 x2 1
 3.42: In Exercises 3944, use long division to divide. 42. 3x4 + x2 1 x2 1
 3.43: In Exercises 3944, use long division to divide. 43. (5x3 13x2 x + 2...
 3.44: In Exercises 3944, use long division to divide. 44. 6x4 + 10x3 + 13...
 3.45: In Exercises 4550, use synthetic division to divide. 45. (3x3 10x2 ...
 3.46: In Exercises 4550, use synthetic division to divide. 46. (2x3 + 6x2...
 3.47: In Exercises 4550, use synthetic division to divide. 47. (0.25x4 4x...
 3.48: In Exercises 4550, use synthetic division to divide. 48. (0.1x3 + 0...
 3.49: In Exercises 4550, use synthetic division to divide. 49. (6x4 4x3 2...
 3.50: In Exercises 4550, use synthetic division to divide. 50. (2x3 + 2x2...
 3.51: In Exercises 51 and 52, use the Remainder Theorem and synthetic div...
 3.52: In Exercises 51 and 52, use the Remainder Theorem and synthetic div...
 3.53: In Exercises 53 and 54, (a) verify the given factor(s) of the funct...
 3.54: In Exercises 53 and 54, (a) verify the given factor(s) of the funct...
 3.55: In Exercises 55 and 56, use the Rational Zero Test to list all poss...
 3.56: In Exercises 55 and 56, use the Rational Zero Test to list all poss...
 3.57: In Exercises 57 and 58, use Descartess Rule of Signs to determine t...
 3.58: In Exercises 57 and 58, use Descartess Rule of Signs to determine t...
 3.59: In Exercises 59 and 60, use synthetic division to verify the upper ...
 3.60: In Exercises 59 and 60, use synthetic division to verify the upper ...
 3.61: In Exercises 6164, find all real zeros of the polynomial function. ...
 3.62: In Exercises 6164, find all real zeros of the polynomial function. ...
 3.63: In Exercises 6164, find all real zeros of the polynomial function. ...
 3.64: In Exercises 6164, find all real zeros of the polynomial function. ...
 3.65: In Exercises 65 68, confirm that the function has the indicated zer...
 3.66: In Exercises 65 68, confirm that the function has the indicated zer...
 3.67: In Exercises 65 68, confirm that the function has the indicated zer...
 3.68: In Exercises 65 68, confirm that the function has the indicated zer...
 3.69: In Exercises 69 and 70, find all the zeros of the function. 69. f(x...
 3.70: In Exercises 69 and 70, find all the zeros of the function.70. f(x)...
 3.71: In Exercises 7176, find all the zeros of the function and write the...
 3.72: In Exercises 7176, find all the zeros of the function and write the...
 3.73: In Exercises 7176, find all the zeros of the function and write the...
 3.74: In Exercises 7176, find all the zeros of the function and write the...
 3.75: In Exercises 7176, find all the zeros of the function and write the...
 3.76: In Exercises 7176, find all the zeros of the function and write the...
 3.77: In Exercises 7782, (a) find all the zeros of the function, (b) writ...
 3.78: In Exercises 7782, (a) find all the zeros of the function, (b) writ...
 3.79: In Exercises 7782, (a) find all the zeros of the function, (b) writ...
 3.80: In Exercises 7782, (a) find all the zeros of the function, (b) writ...
 3.81: In Exercises 7782, (a) find all the zeros of the function, (b) writ...
 3.82: In Exercises 7782, (a) find all the zeros of the function, (b) writ...
 3.83: In Exercises 83 86, find a polynomial function with real coefficien...
 3.84: In Exercises 83 86, find a polynomial function with real coefficien...
 3.85: In Exercises 83 86, find a polynomial function with real coefficien...
 3.86: In Exercises 83 86, find a polynomial function with real coefficien...
 3.87: In Exercises 87 and 88, write the polynomial (a) as the product of ...
 3.88: In Exercises 87 and 88, write the polynomial (a) as the product of ...
 3.89: In Exercises 89 and 90, use the given zero to find all the zeros of...
 3.90: In Exercises 89 and 90, use the given zero to find all the zeros of...
 3.91: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.92: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.93: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.94: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.95: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.96: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.97: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.98: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.99: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.100: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.101: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.102: In Exercises 91102, (a) find the domain of the function, (b) decide...
 3.103: The cost C (in millions of dollars) for the U.S. government to seiz...
 3.104: A biology class performs an experiment comparing the quantity of fo...
 3.105: In Exercises 105108, find all of the vertical, horizontal, and slan...
 3.106: In Exercises 105108, find all of the vertical, horizontal, and slan...
 3.107: In Exercises 105108, find all of the vertical, horizontal, and slan...
 3.108: In Exercises 105108, find all of the vertical, horizontal, and slan...
 3.109: In Exercises 109118, sketch the graph of the rational function by h...
 3.110: In Exercises 109118, sketch the graph of the rational function by h...
 3.111: In Exercises 109118, sketch the graph of the rational function by h...
 3.112: In Exercises 109118, sketch the graph of the rational function by h...
 3.113: In Exercises 109118, sketch the graph of the rational function by h...
 3.114: In Exercises 109118, sketch the graph of the rational function by h...
 3.115: In Exercises 109118, sketch the graph of the rational function by h...
 3.116: In Exercises 109118, sketch the graph of the rational function by h...
 3.117: In Exercises 109118, sketch the graph of the rational function by h...
 3.118: In Exercises 109118, sketch the graph of the rational function by h...
 3.119: A parks and wildlife commission releases 80,000 fish into a lake. A...
 3.120: A page that is x inches wide and y inches high contains 30 square i...
 3.121: In Exercises 121 and 122, determine whether the scatter plot could ...
 3.122: In Exercises 121 and 122, determine whether the scatter plot could ...
 3.123: The table shows the numbers of commercial FM radio stations S in th...
 3.124: In Exercises 124 and 125, determine whether the statement is true o...
 3.125: In Exercises 124 and 125, determine whether the statement is true o...
Solutions for Chapter 3: Polynomial and Rational Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 3: Polynomial and Rational Functions
Get Full SolutionsSince 125 problems in chapter 3: Polynomial and Rational Functions have been answered, more than 61741 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. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3: Polynomial and Rational Functions includes 125 full stepbystep solutions.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.