 81.1: 45m n _3 20 n 7
 81.2: _a + b a 2  b 2
 81.3: x _ 2 + 6x + 9 x + 3
 81.4: 36 c 3 d _2 54 cd
 81.5: STANDARDIZED TEST PRACTICE Identify all values of y for which __y ...
 81.6: 9y 2  6y __3 2y 2 + 5y  12
 81.7: b3  a _3 a2  b2
 81.8: 2 a _ 2 5 b 2c 3b c _ 3 8 a
 81.9: _3t + 6 7t  7 _14t  14 5t + 10
 81.10: 35 16 x 2 _21 4x
 81.11: 20x y _ 3 21 15 x 3 y _ 2 14
 81.12: 12 p __ 2 + 6p  6 4 (p + 1)2 _ 6p  3 2p + 10
 81.13: x _ 2 + 6x + 9 x 2 + 7x + 6 _4x + 12 3x + 3
 81.14: c 3 d _3 _a x c 2 _d a x
 81.15: 2y y _ 2  4 _3 y 2  4y + 4
 81.16: 30bc 12 b 3
 81.17: 3 mn _3 21 m 2 n 2
 81.18: 5t  5 t 2  1
 81.19: c + 5 2c + 10
 81.20: 3t  6 2  t
 81.21: 9  t _ 2 t 2 + t  12
 81.22: 3xyz 4xz 6 x _ 2 3 y 2
 81.23: 4ab 21c 14 c _2 18 a 2
 81.24: 3 5d ( _9 15df)
 81.25: p 3 _ 2q _ p 4q
 81.26: 3 t _ 2 t + 2 _t + 2 t 2
 81.27: 4w + 4 3 _1 w + 1
 81.28: 4t _ 2  4 9 (t + 1)2 _3t + 3 2t  2
 81.29: 3p  21 p 2  49 p _ 2  7p 3p
 81.30: m _ 3 _3n  m _ 4 9 n 2
 81.31: p 3 _ _ 2q  p 2 _ 4q
 81.32: m + n _5 m 2 + n _2 5
 81.33: x + y _ 2x  y _ x + y 2x + y
 81.34: Under what conditions is __x  4 (x + 5)(x  1) undefined?
 81.35: For what values is __2d (d + 1) (d + 1)(d 2  4) undefined?
 81.36: GEOMETRY A parallelogram with an area of 6 x 2  7x  5 square unit...
 81.37: GEOMETRY Parallelogram F has an area of 8 x 2 + 10x  3 square mete...
 81.38: (3 x 2 y) _ 3 9 x 2 y 2
 81.39: (2r s 2) _ 2 12 r 2 s 3
 81.40: (5m n 2) _ 3 5 m 2 n 4
 81.41: y __ 2 + 4y + 4 3 y 2 + 5y  2
 81.42: a __ 2 + 2a + 1 2 a 2 + 3a + 1
 81.43: 3 x __ 2  2x  8 3 x 2  12
 81.44: a _ 2  4 6  3a
 81.45: b _ 2  4b + 3 3  2b  b 2
 81.46: 6 x __2  6 14 x 2  28x + 14
 81.47: 25 a 2 b _3 6 x 2 y 8x y _ 2 20 a 3 b 2
 81.48: _ 9cd 8xw (4w) _2 15c
 81.49: 2 x 3 _y z 5 (_ 4xy z 3 ) 2
 81.50: w __ 2  11w + 24 w 2  18w + 80 w __ 2  15w + 50 w 2  9w + 20
 81.51: r _ 2 + 2r  8 r 2 + 4r + 3 _r  2 3r + 3
 81.52: 5 x __ 2  5x  30 __45  15x 6 + x  x _2 4x  12
 81.53: Under what conditions is a 2 + ab + b _2 a 2  b 2 undefined?
 81.54: Write a ratio to represent the ratio of the number of career field ...
 81.55: Suppose Ray Allen attempted a field goals and made m field goals du...
 81.56: Write a rational expression to represent the ratio of the distance ...
 81.57: Simplify the rational expression. What does this expression tell yo...
 81.58: Under what condition is the rational expression undefined? Describe...
 81.59: Simplify 15 x __2 + 10x 5x . What do you observe about the express...
 81.60: Graph f(x) and g(x) on a graphing calculator. How do the graphs app...
 81.61: Use the table feature to examine the function values for f(x) and g...
 81.62: How can you use what you have observed with f(x) and g(x) to verify...
 81.63: OPEN ENDED Write two rational expressions that are equivalent.
 81.64: CHALLENGE Rewrite _a + b a2 + b so it has a numerator of 1.
 81.65: Which One Doesnt Belong? Identify the expression that does not belo...
 81.66: REASONING Determine whether _2d + 5 3d + 5 = _2 3 is sometimes, alw...
 81.67: REASONING Determine whether _2d + 5 3d + 5 = _2 3 is sometimes, alw...
 81.68: ACT/SAT For what value(s) of x is _4x x 2  x undefined? A 1, 1 B ...
 81.69: REVIEW Which is the simplified form of F 4x _ 7 y4 z5 H _4 y3 z5 G ...
 81.70: y = x 2 71
 81.71: y = x  1 72
 81.72: y = 2 x + 1
 81.73: Determine whether f (x) = x  2 and g (x) = 2x are inverse function...
 81.74: f(x) O x
 81.75: f(x) O x
 81.76: (x) O x
 81.77: ASTRONOMY Earth is an average of 1.496 10 8 kilometers from the Sun...
 81.78: r 2  3r = 4 7
 81.79: 18 u 2  3u = 1 8
 81.80: d 2  5d = 0
 81.81: 2x + 7 + 5 = 0 8
 81.82: 5 3x  4 = x + 1
 81.83: 2 3 + x =  _4 9
 81.84: x + _5 8 =  _5 6
 81.85: x  _3 5 = _2 3
 81.86: x + _3 16 =  _1 2
 81.87: x  _1 6 =  _7 9
 81.88: x  _3 8
Solutions for Chapter 81: Multiplying and Dividing Rational Expressions
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 81: Multiplying and Dividing Rational Expressions
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 81: Multiplying and Dividing Rational Expressions includes 88 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Since 88 problems in chapter 81: Multiplying and Dividing Rational Expressions have been answered, more than 56404 students have viewed full stepbystep solutions from this chapter.

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.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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].

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.

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.

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].

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Outer product uv T
= column times row = rank one matrix.

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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 ().

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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