 6.3.1: A rational expression whose numerator or denominator or both contai...
 6.3.2: 7 x + 5 x2 5 x + 1 = x2 x2 # a 7 x + 5 x2 b a 5 x + 1b = x2 # 7 x +...
 6.3.3: 1 x + 3  1 x 3 = x1x + 32 x1x + 32 # a 1 x + 3  1 x b 3 = x(x + 3...
 6.3.4: x 5  5 x 1 5 + 1 x
 6.3.5: 1 x + 1 y 1 x  1 y
 6.3.6: x y + 1 x y x + 1 x
 6.3.7: 8x2  2x1 10x1  6x2
 6.3.8: 12x2  3x1 15x1  9x2
 6.3.9: 1 x  2 1  1 x  2
 6.3.10: 1 x + 2 1 + 1 x + 2
 6.3.11: 1 x + 5  1 x 5
 6.3.12: 1 x + 6  1 x 6
 6.3.13: 4 x + 4 1 x + 4  1 x
 6.3.14: 7 x + 7 1 x + 7  1 x
 6.3.15: 1 x  1 + 1 1 x + 1  1
 6.3.16: 1 x + 1  1 1 x  1 + 1
 6.3.17: x1 + y1 (x + y) 1
 6.3.18: (x1 + y1 ) 1
 6.3.19: x + 2 x  2  x  2 x + 2 x  2 x + 2 + x + 2 x  2
 6.3.20: x + 1 x  1  x  1 x + 1 x  1 x + 1 + x + 2 x  1
 6.3.21: 2 x3y + 5 xy4 5 x3y  3 xy
 6.3.22: 3 xy2 + 2 x2y 1 x2y + 2 xy3
 6.3.23: 3 x + 2  3 x  2 5 x2  4
 6.3.24: 3 x + 1  3 x  1 5 x2  1
 6.3.25: 3a1 + 3b1 4a2  9b2
 6.3.26: 5a1  2b1 25a2  4b2
 6.3.27: 4x x2  4  5 x  2 2 x  2 + 3 x + 2
 6.3.28: 2 x + 3 + 5x x2  9 4 x + 3 + 2 x  3
 6.3.29: 2y y2 + 4y + 3 1 y + 3 + 2 y + 1
 6.3.30: 5y y2  5y + 6 3 y  3 + 2 y  2
 6.3.31: 2 a2  1 ab  1 b2 1 a2  3 ab + 2 b2
 6.3.32: 2 b2  5 ab  3 a2 2 b2 + 7 ab + 3 a2
 6.3.33: 2x x2  25 + 1 3x  15 5 x  5 + 3 4x  20
 6.3.34: 7x 2x  2 + x x2  1 4 x + 1  1 3x + 3
 6.3.35: 3 x + 2y  2y x2 + 2xy 3y x2 + 2xy + 5 x
 6.3.36: 1 x3  y3 1 x  y  1 x2 + xy + y2
 6.3.37: 2 m2  3m + 2 + 2 m2  m  2 2 m2  1 + 2 m2 + 4m + 3
 6.3.38: m m2  9  2 m2  4m + 4 3 m2  5m + 6 + m m2 + m  6
 6.3.39: 2 a2 + 2a  8 + 1 a2 + 5a + 4 1 a2  5a + 6 + 2 a2  a  2
 6.3.40: 3 a2 + 10a + 25  1 a2  a  2 4 a2 + 6a + 5  2 a2 + 3a  10
 6.3.41: x  1 x2  4 1 + 1 x  2  1 x  2
 6.3.42: x  3 x2  16 1 + 1 x  4  1 x  4
 6.3.43: 3 1  3 3 + x  3 3 3  x  1
 6.3.44: 5 1  5 5 + x  5 5 5  x  1
 6.3.45: x 1  1 1 + 1 x
 6.3.46: 1 x + 1 x  1 x + 1 x
 6.3.47: Find f a 1 x + 3 b and simplify.
 6.3.48: Find f a 1 x  6 b and simplify
 6.3.49: f(x) = 3 x
 6.3.50: f(x) = 1 x2
 6.3.51: How much are your monthly payments on a loan? If P is the principal...
 6.3.52: The average rate on a roundtrip commute having a oneway distance ...
 6.3.53: If three resistors with resistances R1 , R2 , and R3 are connected ...
 6.3.54: A camera lens has a measurement called its focal length, f. When an...
 6.3.55: What is a complex rational expression? Give an example with your ex...
 6.3.56: Describe two ways to simplify 2 x + 2 y 2 x  2 y .
 6.3.57: Which method do you prefer for simplifying complex rational express...
 6.3.58: Of the four complex rational expressions in Exercises 5154, which o...
 6.3.59: x  1 2x + 1 1  x 2x + 1 = 2x  1
 6.3.60: 1 x + 1 1 x = 2
 6.3.61: 1 x + 1 3 1 3x = x + 1 3
 6.3.62: x 3 2 x + 1 = 3(x + 1) 2
 6.3.63: I simplified 1 + 3x xy 5 + 4y by multiplying the numerator by xy.
 6.3.64: By noticing that 1 x + 7  1 x 7 repeats x and 7 twice, its fairly ...
 6.3.65: I simplified 3  6 x + 5 1 + 7 x  4 by multiplying by 1 and obtain...
 6.3.66: Before simplifying 1  x2 1  5x3 , I wrote the complex fraction ...
 6.3.67: Simplify: x + h x + h + 1  x x + 1 h
 6.3.68: Simplify: x + 1 x + 1 x + 1 x
 6.3.69: If f(x) = 1 x + 1 , find f(f(a)) and simplify
 6.3.70: Let x represent the first of two consecutive integers. Find a simpl...
 6.3.71: Solve: x2 + 27 = 12x. (Section 5.7, Example 2)
 6.3.72: Multiply: (4x2  y) 2 . (Section 5.2, Example 7)
 6.3.73: Solve: 4 6 3x  7 6 8. (Section 4.2, Example 4)
 6.3.74: Simplify: 8x4y5 4x3y2 .
 6.3.75: Divide 737 by 21 without using a calculator. Write the answer as qu...
 6.3.76: Simplify: 6x2 + 3x  (6x2  4x).
Solutions for Chapter 6.3: Complex Rational Expressions
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6.3: Complex Rational Expressions
Get Full SolutionsChapter 6.3: Complex Rational Expressions includes 76 full stepbystep solutions. Since 76 problems in chapter 6.3: Complex Rational Expressions have been answered, more than 53100 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

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

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.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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}.

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.