 P.6.1: In Exercises 16, find all numbers that must be excluded from the d...
 P.6.2: In Exercises 16, find all numbers that must be excluded from the d...
 P.6.3: In Exercises 16, find all numbers that must be excluded from the d...
 P.6.4: In Exercises 16, find all numbers that must be excluded from the d...
 P.6.5: In Exercises 16, find all numbers that must be excluded from the d...
 P.6.6: In Exercises 16, find all numbers that must be excluded from the d...
 P.6.7: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.8: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.9: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.10: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.11: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.12: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.13: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.14: In Exercises 714, simplify each rational exprrs:sion. Find af/ IIW...
 P.6.15: In Exercises 1532. multiply or divide as indicated. r~ 2.r~4
 P.6.16: In Exercises 1532. multiply or divide as indicated. 6.t+9 x  5 lt...
 P.6.17: In Exercises 1532. multiply or divide as indicated. .<':.9. x'  3...
 P.6.18: In Exercises 1532. multiply or divide as indicated. x2 4 2x  4 x...
 P.6.19: In Exercises 1532. multiply or divide as indicated. x'Sx  6 x' ...
 P.6.20: In Exercises 1532. multiply or divide as indicated. x1+5x+6 x1  9...
 P.6.21: In Exercises 1532. multiply or divide as indicated. rl gxT? , . ...
 P.6.22: In Exercises 1532. multiply or divide as indicated. t' + t>r + 9 ~...
 P.6.23: In Exercises 1532. multiply or divide as indicated. tl lrr3
 P.6.24: In Exercises 1532. multiply or divide as indicated. x5. ~x 202A. ...
 P.6.25: In Exercises 1532. multiply or divide as indicated. x' 4 x  2X x...
 P.6.26: In Exercises 1532. multiply or divide as indicated. xl4 x+2.r  2...
 P.6.27: In Exercises 1532. multiply or divide as indicated. 4.r + 10 6x' +...
 P.6.28: In Exercises 1532. multiply or divide as indicated. t2 x x2  1 21...
 P.6.29: In Exercises 1532. multiply or divide as indicated. x1  25 x' + I...
 P.6.30: In Exercises 1532. multiply or divide as indicated. .r .t xl+5.t+...
 P.6.31: In Exercises 1532. multiply or divide as indicated. x'!  x  12 x...
 P.6.32: In Exercises 1532. multiply or divide as indicated. .r> 25x ' 4.r...
 P.6.33: In Exercises 3358, add or subtract as indicated. 4.r+1+8.r ~ 6r + ...
 P.6.34: In Exercises 3358, add or subtract as indicated. 3x 2 3x~6 34' :;...
 P.6.35: In Exercises 3358, add or subtract as indicated. cl 2x x J..,_. ...
 P.6.36: In Exercises 3358, add or subtract as indicated. x'4.r 4.r ~ .16 ...
 P.6.37: In Exercises 3358, add or subtract as indicated. 4,t  10 X  4.r...
 P.6.38: In Exercises 3358, add or subtract as indicated. ~  2...::...!.....
 P.6.39: In Exercises 3358, add or subtract as indicated. ,\'1 + 3x39' ,tT...
 P.6.40: In Exercises 3358, add or subtract as indicated. :r2  lt 40 .;; ...
 P.6.41: In Exercises 3358, add or subtract as indicated. 3 6 +  .r 4 x+ S
 P.6.42: In Exercises 3358, add or subtract as indicated. 8 + _ 2_ x  2 x  3
 P.6.43: In Exercises 3358, add or subtract as indicated. 3 3  X+ 1 X
 P.6.44: In Exercises 3358, add or subtract as indicated. 4 3  X x3
 P.6.45: In Exercises 3358, add or subtract as indicated. ~ + !..!.1.r+2 x  2
 P.6.46: In Exercises 3358, add or subtract as indicated. 1:!:....  !..!.....
 P.6.47: In Exercises 3358, add or subtract as indicated. t + 5 _ .r  ~.r...
 P.6.48: In Exercises 3358, add or subtract as indicated. x .,.. J x  3 ...
 P.6.49: In Exercises 3358, add or subtract as indicated. 3 1 , T 2t+4 3x 6
 P.6.50: In Exercises 3358, add or subtract as indicated. 5 71x  8 + 3x  12
 P.6.51: In Exercises 3358, add or subtract as indicated. 4 4 +  x' +6x +...
 P.6.52: In Exercises 3358, add or subtract as indicated. 3 5.t .+ 5xT2 2...
 P.6.53: In Exercises 3358, add or subtract as indicated. X _ X"x2 + 3x  1...
 P.6.54: In Exercises 3358, add or subtract as indicated. X _ X"x1  2r  2...
 P.6.55: In Exercises 3358, add or subtract as indicated. x+3 x+2   x2...
 P.6.56: In Exercises 3358, add or subtract as indicated. x + 5 _ x + 1 ~x2...
 P.6.57: In Exercises 3358, add or subtract as indicated. 4.t2 +x  6 3x 5...
 P.6.58: In Exercises 3358, add or subtract as indicated. 6x2 +17x  40 3 S...
 P.6.59: In Exercises 5972, simplify each complex rational expression. :!: ...
 P.6.60: In Exercises 5972, simplify each complex rational expression. !. ...
 P.6.61: In Exercises 5972, simplify each complex rational expression. S + ...
 P.6.62: In Exercises 5972, simplify each complex rational expression. S + ...
 P.6.63: In Exercises 5972, simplify each complex rational expression. I 1 ...
 P.6.64: In Exercises 5972, simplify each complex rational expression. I1 ...
 P.6.65: In Exercises 5972, simplify each complex rational expression. x  ...
 P.6.66: In Exercises 5972, simplify each complex rational expression. x  ...
 P.6.67: In Exercises 5972, simplify each complex rational expression. 3 47...
 P.6.68: In Exercises 5972, simplify each complex rational expression. X ...
 P.6.69: In Exercises 5972, simplify each complex rational expression. x+1...
 P.6.70: In Exercises 5972, simplify each complex rational expression. :=...
 P.6.71: In Exercises 5972, simplify each complex rational expression. .t +...
 P.6.72: In Exercises 5972, simplify each complex rational expression. X+ I...
 P.6.73: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.74: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.75: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.76: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.77: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.78: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.79: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.80: In Exercises 7380, perform the indicated operations. Simplify the ...
 P.6.81: The rational expression 130x 100 X desc.ribcs the oost, in millions...
 P.6.82: The average rate on a roundtrip commute having a oneway distanced...
 P.6.83: The bar graph shO\ the estimated number of calories per day needed ...
 P.6.84: In Exercises 8485, express the perimeter of each rectangle as a si...
 P.6.85: In Exercises 8485, express the perimeter of each rectangle as a si...
 P.6.86: Whal is a rational expression?
 P.6.87: Explain how to determine which numbe rs must be excluded from the d...
 P.6.88: Explain how to simplify a rational expression.
 P.6.89: Explain how to multiply rational expressions.
 P.6.90: Explain how to divide rational expressions.
 P.6.91: Explain how to add or subtract rational expressions with the same d...
 P.6.92: Explain how to add rational exp e~ ons having no common ' 7 factors...
 P.6.93: Explain how to find the least common denominator ror denominators o...
 P.6.94: Describe two ways to simplify .\; x~ . ; +  x X
 P.6.95: Explain the error in Exercises 9597. Then rewrite the right side o...
 P.6.96: Explain the error in Exercises 9597. Then rewrite the right side o...
 P.6.97: Explain the error in Exercises 9597. Then rewrite the right side o...
 P.6.98: In Exercises 98101, determine whether each statement makes sense o...
 P.6.99: In Exercises 98101, determine whether each statement makes sense o...
 P.6.100: In Exercises 98101, determine whether each statement makes sense o...
 P.6.101: I subtracted 3x 5 rom and obtained a constant x  1 x  1
 P.6.102: In Exercises 102105, determine whether each statement is true or f...
 P.6.103: In Exercises 102105, determine whether each statement is true or f...
 P.6.104: In Exercises 102105, determine whether each statement is true or f...
 P.6.105: In Exercises 102105, determine whether each statement is true or f...
 P.6.106: In Exercises 1~108.perform the indicated operations. 1   o:''...
 P.6.107: In Exercises 1~108.perform the indicated operations. (1:)(1  1 )...
 P.6.108: In Exercises 1~108.perform the indicated operations. (x  .vr' + (x...
 P.6.109: In one short sente nce, five words or less, explain what 1 I I  + ...
 P.6.110: Exercises 110112 will help you prepare for the material c.overed ...
 P.6.111: Exercises 110112 will help you prepare for the material c.overed ...
 P.6.112: Exercises 110112 will help you prepare for the material c.overed ...
Solutions for Chapter P.6: Rational Expressions
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter P.6: Rational Expressions
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780321782281. This textbook survival guide was created for the textbook: College Algebra , edition: 6. Since 112 problems in chapter P.6: Rational Expressions have been answered, more than 35976 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter P.6: Rational Expressions includes 112 full stepbystep solutions.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.