 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 22544 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.

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.

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

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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).

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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 ().

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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