 3.5.1: Functions of the form f(x) = N(x)D(x), where N(x) and D(x) are poly...
 3.5.2: If f(x) as xa from the left (or right), then x = a is a _______ of ...
 3.5.3: What feature of the graph of y = 9 x 3 can you find by solving x 3 ...
 3.5.4: Is y = 2 3 a horizontal asymptote of the function f(x) = 2x 3x2 5
 3.5.5: In Exercises 510, (a) find the domain of the function, (b) complete...
 3.5.6: In Exercises 510, (a) find the domain of the function, (b) complete...
 3.5.7: In Exercises 510, (a) find the domain of the function, (b) complete...
 3.5.8: In Exercises 510, (a) find the domain of the function, (b) complete...
 3.5.9: In Exercises 510, (a) find the domain of the function, (b) complete...
 3.5.10: In Exercises 510, (a) find the domain of the function, (b) complete...
 3.5.11: In Exercises 1116, match the function with its graph. [The graphs a...
 3.5.12: In Exercises 1116, match the function with its graph. [The graphs a...
 3.5.13: In Exercises 1116, match the function with its graph. [The graphs a...
 3.5.14: In Exercises 1116, match the function with its graph. [The graphs a...
 3.5.15: In Exercises 1116, match the function with its graph. [The graphs a...
 3.5.16: In Exercises 1116, match the function with its graph. [The graphs a...
 3.5.17: In Exercises 1720, find any asymptotes of the graph of the rational...
 3.5.18: In Exercises 1720, find any asymptotes of the graph of the rational...
 3.5.19: In Exercises 1720, find any asymptotes of the graph of the rational...
 3.5.20: In Exercises 1720, find any asymptotes of the graph of the rational...
 3.5.21: In Exercises 2124, find any asymptotes and holes in the graph of th...
 3.5.22: In Exercises 2124, find any asymptotes and holes in the graph of th...
 3.5.23: In Exercises 2124, find any asymptotes and holes in the graph of th...
 3.5.24: In Exercises 2124, find any asymptotes and holes in the graph of th...
 3.5.25: In Exercises 2528, (a) find the domain of the function, (b) decide ...
 3.5.26: In Exercises 2528, (a) find the domain of the function, (b) decide ...
 3.5.27: In Exercises 2528, (a) find the domain of the function, (b) decide ...
 3.5.28: In Exercises 2528, (a) find the domain of the function, (b) decide ...
 3.5.29: In Exercises 2932, (a) determine the domains of f and g, (b) find a...
 3.5.30: In Exercises 2932, (a) determine the domains of f and g, (b) find a...
 3.5.31: In Exercises 2932, (a) determine the domains of f and g, (b) find a...
 3.5.32: In Exercises 2932, (a) determine the domains of f and g, (b) find a...
 3.5.33: In Exercises 3336, determine the value that the function f approach...
 3.5.34: In Exercises 3336, determine the value that the function f approach...
 3.5.35: In Exercises 3336, determine the value that the function f approach...
 3.5.36: In Exercises 3336, determine the value that the function f approach...
 3.5.37: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.38: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.39: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.40: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.41: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.42: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.43: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.44: In Exercises 3744, find the zeros (if any) of the rational function...
 3.5.45: The game commission introduces 100 deer into newly acquired state g...
 3.5.46: The endpoints of the interval over which distinct vision is possibl...
 3.5.47: Consider a physics laboratory experiment designed to determine an u...
 3.5.48: The sales S (in thousands of units) of a tablet computer during the...
 3.5.49: The cost C (in dollars) of supplying recycling bins to p% of the po...
 3.5.50: In Exercises 50 and 51, determine whether the statement is true or ...
 3.5.51: In Exercises 50 and 51, determine whether the statement is true or ...
 3.5.52: Write a rational function f that has the specified characteristics....
 3.5.53: A real zero of the numerator of a rational function f is x = c. Mus...
 3.5.54: When the graph of a rational function f has a vertical asymptote at...
 3.5.55: Use a graphing utility to compare the graphs of y1 and y2. y1 = 3x3...
 3.5.56: The graph of a rational function f(x) = N(x) D(x) is shown below. D...
 3.5.57: In Exercises 57 60, write the general form of the equation of the l...
 3.5.58: In Exercises 57 60, write the general form of the equation of the l...
 3.5.59: In Exercises 57 60, write the general form of the equation of the l...
 3.5.60: In Exercises 57 60, write the general form of the equation of the l...
 3.5.61: In Exercises 61 64, divide using long division. 61. (x2 + 5x + 6) (...
 3.5.62: In Exercises 61 64, divide using long division.62. (x2 10x + 15) (x 3)
 3.5.63: In Exercises 61 64, divide using long division.63. (2x4 + x2 11) (x...
 3.5.64: In Exercises 61 64, divide using long division. 64. (4x5 + 3x3 10) ...
Solutions for Chapter 3.5: Polynomial and Rational Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 3.5: Polynomial and Rational Functions
Get Full SolutionsChapter 3.5: Polynomial and Rational Functions includes 64 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 64 problems in chapter 3.5: Polynomial and Rational Functions have been answered, more than 60803 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.