 3.1: In Exercises 14. use the vertex and illlercepts to sketch the grap...
 3.2: In Exercises 14. use the vertex and illlercepts to sketch the grap...
 3.3: In Exercises 14. use the vertex and illlercepts to sketch the grap...
 3.4: In Exercises 14. use the vertex and illlercepts to sketch the grap...
 3.5: In Exercises 56, use the function's equation, and not its gmph. to...
 3.6: In Exercises 56, use the function's equation, and not its gmph. to...
 3.7: A quarte rback tosses a football to a receiver 40 yards downfield. ...
 3.8: A field bordering a straight stream is to be enclosed. The side bor...
 3.9: Among all pairs or numbers whose difference is 14, find a pair whos...
 3.10: In Exercises 1013, use the Leading Coefficient Test to determine ...
 3.11: In Exercises 1013, use the Leading Coefficient Test to determine ...
 3.12: In Exercises 1013, use the Leading Coefficient Test to determine ...
 3.13: In Exercises 1013, use the Leading Coefficient Test to determine ...
 3.14: The Brazilian Amazon rain forest is the wortds largest tropa1 rain ...
 3.15: lbe polynomial function /{x)  {).87..' + 0.35.r' + 81.62.< + 76!...
 3.16: In Exercises 1617, find the zeros for each polynomial function and...
 3.17: In Exercises 1617, find the zeros for each polynomial function and...
 3.18: Show that j{x)  x'  2x  I h:u a real zero between and2.
 3.19: In Exercises 1924. a. Use the Leading Coefficient Test to d<t.rmin...
 3.20: In Exercises 1924. a. Use the Leading Coefficient Test to d<t.rmin...
 3.21: In Exercises 1924. a. Use the Leading Coefficient Test to d<t.rmin...
 3.22: In Exercises 1924. a. Use the Leading Coefficient Test to d<t.rmin...
 3.23: In Exercises 1924. a. Use the Leading Coefficient Test to d<t.rmin...
 3.24: In Exercises 1924. a. Use the Leading Coefficient Test to d<t.rmin...
 3.25: In Exercises 2526. graph each polynomial function. j{.r)  lr(x  ...
 3.26: In Exercises 2526. graph each polynomial function. f(.r)   x'(x ...
 3.27: In Exercises 2729, divide using long lfivision. (4x' 3x'  lt + 1...
 3.28: In Exercises 2729, divide using long lfivision. (lOx'  26.t2 + 17...
 3.29: In Exercises 2729, divide using long lfivision. (4.<' + 6.r' + 3x ...
 3.30: In Exercises 3031, divide using synthetic division. (lx' ~ 11..' ...
 3.31: In Exercises 3031, divide using synthetic division. (lr'  2.r'  ...
 3.32: Given j{x)  1x'  7>2 + 9x  3. usc the Remainder 111corem to find...
 3.33: Use synthetic division to divide j{.t)  2.r' + x'  13x 6 by x  2...
 3.34: Solve the equation x'  17x + 4  0 given that 4 is a root.
 3.35: In Exercises 3536, us~ the Rational z~ro Theorem to list alf poJSi...
 3.36: In Exercises 3536, us~ the Rational z~ro Theorem to list alf poJSi...
 3.37: In Exercises 3738, use Descartes's Ruft! of Signs to determine the...
 3.38: In Exercises 3738, use Descartes's Ruft! of Signs to determine the...
 3.39: Use Descartes's Rule of Signs to explain why 2.' + 6.r' + 8  0 ha...
 3.40: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.41: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.42: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.43: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.44: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.45: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.46: For Exercises 4046. a. Lift all possiblt rational TOOlS or rationa...
 3.47: In Exercises 4748, find an nihdegree polynomial function with rea...
 3.48: In Exercises 4748, find an nihdegree polynomial function with rea...
 3.49: In Exercises 4950, find all the zeros of each polynomial function ...
 3.50: In Exercises 4950, find all the zeros of each polynomial function ...
 3.51: ln Exercises 51 54, graphs of fiftltdegr.e polytlomia/ funclwns ar...
 3.52: ln Exercises 51 54, graphs of fiftltdegr.e polytlomia/ funclwns ar...
 3.53: ln Exercises 51 54, graphs of fiftltdegr.e polytlomia/ funclwns ar...
 3.54: ln Exercises 51 54, graphs of fiftltdegr.e polytlomia/ funclwns ar...
 3.55: In Exercises 55 56, use transformations of f(x)  .!._or f(x)  ~ ...
 3.56: In Exercises 55 56, use transformations of f(x)  .!._or f(x)  ~ ...
 3.57: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.58: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.59: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.60: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.61: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.62: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.63: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.64: In Exercises 5764, find the vertical asymptotes, if ally, lite hor...
 3.65: A company is planning to manufacture affordable graphing calculator...
 3.66: Exercises 6667 involve rational functions that model the given sil...
 3.67: Exercises 6667 involve rational functions that model the given sil...
 3.68: The bar graph shows the population of lhe United Slates, in million...
 3.69: In Exercises 6974, solve each inequality and graph the solution se...
 3.70: In Exercises 6974, solve each inequality and graph the solution se...
 3.71: In Exercises 6974, solve each inequality and graph the solution se...
 3.72: In Exercises 6974, solve each inequality and graph the solution se...
 3.73: In Exercises 6974, solve each inequality and graph the solution se...
 3.74: In Exercises 6974, solve each inequality and graph the solution se...
 3.75: The graph shows stopping distances for motorcycles at various speed...
 3.76: Use the position function s(l)   1612 + vrJ + s0 to solve this pr...
 3.77: Solve the variation problems in Exercises 7782. Many areas of Nort...
 3.78: Solve the variation problems in Exercises 7782. The distance that ...
 3.79: Solve the variation problems in Exercises 7782. The pitch of a mus...
 3.80: Solve the variation problems in Exercises 7782. The loudness of a ...
 3.81: Solve the variation problems in Exercises 7782.1l1e time required ...
 3.82: Solve the variation problems in Exercises 7782. The volume of a py...
 3.83: Heart rates and life spans o r most mammals c.an be modeled using i...
Solutions for Chapter 3: Polynomial and Rational Functions
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 3: Polynomial and Rational Functions
Get Full SolutionsChapter 3: Polynomial and Rational Functions includes 83 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 6. College Algebra was written by and is associated to the ISBN: 9780321782281. Since 83 problems in chapter 3: Polynomial and Rational Functions have been answered, more than 24858 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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