 6.1: If f(x) = x2 + 2x  3 x2  4 , find the following function values. ...
 6.2: f(x) = x  6 (x  3)(x + 4)
 6.3: f(x) = x + 2 x2 + x  2
 6.4: 5x3  35x 15x2
 6.5: x2 + 6x  7 x2  49
 6.6: 6x2 + 7x + 2 2x2  9x  5
 6.7: x2 + 4 x2  4
 6.8: x3  8 x2  4
 6.9: 5x2  5 3x + 12 # x + 4 x  1
 6.10: 2x + 5 4x2 + 8x  5 # 4x2  4x + 1 x + 1
 6.11: x2  9x + 14 x3 + 2x2 # x2  4 x2  4x + 4
 6.12: 1 x2 + 8x + 15 , 3 x + 5
 6.13: x2 + 16x + 64 2x2  128 , x2 + 10x + 16 x2  6x  16
 6.14: y2  16 y3  64 , y2  3y  18 y2 + 5y + 6
 6.15: x2  4x + 4  y2 2x2  11x + 15 # x4y x  2 + y , x3y  2x2y  x2y2...
 6.16: Deer are placed into a newly acquired habitat. The deer population ...
 6.17: 4x + 1 3x  1 + 8x  5 3x  1
 6.18: 2x  7 x2  9  x  4 x2  9
 6.19: 4x2  11x + 4 x  3  x2  4x + 10 x  3
 6.20: 7 9x3 and 5 12x
 6.21: x + 7 x2 + 2x  35 and x x2 + 9x + 14
 6.22: 1 x + 2 x  5
 6.23: 2 x2  5x + 6 + 3 x2  x  6
 6.24: x  3 x2  8x + 15 + x + 2 x2  x  6
 6.25: 3x2 9x2  16  x 3x + 4
 6.26: y y2 + 5y + 6  2 y2 + 3y + 2
 6.27: x x + 3 + x x  3  9 x2  9
 6.28: 3x2 x  y + 3y2 y  x
 6.29: 3 x  3 8 x  8
 6.30: 5 x + 1 1  25 x2
 6.31: 3  1 x + 3 3 + 1 x + 3
 6.32: 4 x + 3 2 x  2  1 x2 + x  6
 6.33: 2 x2  x  6 + 1 x2  4x + 3 3 x2 + x  2  2 x2 + 5x + 6
 6.34: x 2 + x 1 x 2  x 1
 6.35: (15x3  30x2 + 10x  2) , (5x2 )
 6.36: (36x4y3 + 12x2y3  60x2y2 ) , (6xy2 )
 6.37: (6x2  5x + 5) , (2x + 3)
 6.38: (10x3  26x2 + 17x  13) , (5x  3)
 6.39: (x6 + 3x5  2x4 + x2  3x + 2) , (x  2)
 6.40: (4x4 + 6x3 + 3x  1) , (2x2 + 1)
 6.41: (4x3  3x2  2x + 1) , (x + 1)
 6.42: (3x4  2x2  10x  20) , (x  2)
 6.43: (x4 + 16) , (x + 4)
 6.44: f(x) = 2x3  5x2 + 4x  1; f(2)
 6.45: f(x) = 3x4 + 7x3 + 8x2 + 2x + 4; f 11 3 2
 6.46: 2x3  x2  8x + 4 = 0; 2
 6.47: x4  x3  7x2 + x + 6 = 0; 4
 6.48: Use synthetic division to show that 1 2 is a solution of 6x3 + x2 ...
 6.49: 3 x + 1 3 = 5 x
 6.50: 5 3x + 4 = 3 2x  8
 6.51: 1 x  5  3 x + 5 = 6 x2  25
 6.52: x + 5 x + 1  x x + 2 = 4x + 1 x2 + 3x + 2
 6.53: 2 3  5 3x = 1 x2
 6.54: 2 x  1 = 1 4 + 7 x + 2
 6.55: 2x + 7 x + 5  x  8 x  4 = x + 18 x2 + x  20
 6.56: The function f(x) = 4x 100  x models the cost, f(x), in millions o...
 6.57: P = R  C n for C
 6.58: P1V1 T1 = P2V2 T2 for T1
 6.59: T = A  P Pr for P
 6.60: 1 R = 1 R1 + 1 R2 for R
 6.61: I = nE R + nr for n
 6.62: A company is planning to manufacture affordable graphing calculator...
 6.63: After riding at a steady rate for 60 miles, a bicyclist had a flat ...
 6.64: The current of a river moves at 3 miles per hour. It takes a boat a...
 6.65: Working alone, two people can clean their house in 3 hours and 6 ho...
 6.66: Working together, two crews can clear snow from the citys streets i...
 6.67: An inlet faucet can fill a small pond in 60 minutes. The pond can b...
 6.68: A companys profit varies directly as the number of products it sell...
 6.69: The distance that a body falls from rest varies directly as the squ...
 6.70: The pitch of a musical tone varies inversely as its wavelength. A t...
 6.71: The loudness of a stereo speaker, measured in decibels, varies inve...
 6.72: The time required to assemble computers varies directly as the numb...
 6.73: The volume of a pyramid varies jointly as its height and the area o...
Solutions for Chapter 6: Rational Expressions, Functions, and Equations
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6: Rational Expressions, Functions, and Equations
Get Full SolutionsChapter 6: Rational Expressions, Functions, and Equations includes 73 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 73 problems in chapter 6: Rational Expressions, Functions, and Equations have been answered, more than 89564 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

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).

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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.

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.

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].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

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.

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

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).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.