 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 28854 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.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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