 P.5.1: VOCABULARY: Fill in the blanks.The set of real numbers for which an...
 P.5.2: VOCABULARY: Fill in the blanks.The quotient of two algebraic expres...
 P.5.3: VOCABULARY: Fill in the blanks.Fractional expressions with separate...
 P.5.4: VOCABULARY: Fill in the blanks.To simplify an expression with negat...
 P.5.5: VOCABULARY: Fill in the blanks.Two algebraic expressions that have ...
 P.5.6: VOCABULARY: Fill in the blanks.An important rational expression, su...
 P.5.7: In Exercises 722, find the domain of the expression.3x2 4x 7
 P.5.8: In Exercises 722, find the domain of the expression.2x2 5x 2
 P.5.9: In Exercises 722, find the domain of the expression.4x3 3, x 0
 P.5.10: In Exercises 722, find the domain of the expression.6x2 9, x > 0
 P.5.11: In Exercises 722, find the domain of the expression.13 x
 P.5.12: In Exercises 722, find the domain of the expression.x 63x 2
 P.5.13: In Exercises 722, find the domain of the expression.x2 1x2 2x 1
 P.5.14: In Exercises 722, find the domain of the expression.x2 5x 6x2 4
 P.5.15: In Exercises 722, find the domain of the expression.x2 2x 3x2 6x 9
 P.5.16: In Exercises 722, find the domain of the expression.x2 x 12x2 8x 16
 P.5.17: In Exercises 722, find the domain of the expression.x 7
 P.5.18: In Exercises 722, find the domain of the expression.4 x
 P.5.19: In Exercises 722, find the domain of the expression.2x 5
 P.5.20: In Exercises 722, find the domain of the expression.4x 5
 P.5.21: In Exercises 722, find the domain of the expression.1x 3
 P.5.22: In Exercises 722, find the domain of the expression.1x 2
 P.5.23: In Exercises 23 and 24, find the missing factor in the numeratorsuc...
 P.5.24: In Exercises 23 and 24, find the missing factor in the numeratorsuc...
 P.5.25: In Exercises 2542, write the rational expression in simplestform.15...
 P.5.26: In Exercises 2542, write the rational expression in simplestform.18...
 P.5.27: In Exercises 2542, write the rational expression in simplestform.3x...
 P.5.28: In Exercises 2542, write the rational expression in simplestform.2x...
 P.5.29: In Exercises 2542, write the rational expression in simplestform.4y...
 P.5.30: In Exercises 2542, write the rational expression in simplestform.9x...
 P.5.31: In Exercises 2542, write the rational expression in simplestform.x ...
 P.5.32: In Exercises 2542, write the rational expression in simplestform.12...
 P.5.33: In Exercises 2542, write the rational expression in simplestform.y2...
 P.5.34: In Exercises 2542, write the rational expression in simplestform.x ...
 P.5.35: In Exercises 2542, write the rational expression in simplestform.x ...
 P.5.36: In Exercises 2542, write the rational expression in simplestform.x ...
 P.5.37: In Exercises 2542, write the rational expression in simplestform.y ...
 P.5.38: In Exercises 2542, write the rational expression in simplestform.x2...
 P.5.39: In Exercises 2542, write the rational expression in simplestform.2 ...
 P.5.40: In Exercises 2542, write the rational expression in simplestform.x ...
 P.5.41: In Exercises 2542, write the rational expression in simplestform.z ...
 P.5.42: In Exercises 2542, write the rational expression in simplestform.y ...
 P.5.43: ERROR ANALYSIS Describe the error.43
 P.5.44: ERROR ANALYSIS Describe the error.44
 P.5.45: In Exercises 45 and 46, complete the table. What can youconclude?45
 P.5.46: In Exercises 45 and 46, complete the table. What can youconclude?46
 P.5.47: GEOMETRY In Exercises 47 and 48, find the ratio of thearea of the s...
 P.5.48: GEOMETRY In Exercises 47 and 48, find the ratio of thearea of the s...
 P.5.49: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.50: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.51: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.52: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.53: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.54: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.55: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.56: In Exercises 4956, perform the multiplication or divisionand simpli...
 P.5.57: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.58: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.59: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.60: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.61: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.62: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.63: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.64: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.65: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.66: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.67: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.68: In Exercises 5768, perform the addition or subtraction andsimplify....
 P.5.69: ERROR ANALYSIS In Exercises 69 and 70, describe theerror.69
 P.5.70: ERROR ANALYSIS In Exercises 69 and 70, describe theerror.70
 P.5.71: In Exercises 7176, simplify the complex fraction.x2 1x 2
 P.5.72: In Exercises 7176, simplify the complex fraction.x 4x4 4x
 P.5.73: In Exercises 7176, simplify the complex fraction. x2x 12 xx 13
 P.5.74: In Exercises 7176, simplify the complex fraction.x2 1x x 12x
 P.5.75: In Exercises 7176, simplify the complex fraction.x 12xx
 P.5.76: In Exercises 7176, simplify the complex fraction. t 2t 2 1 t 2 1t 2
 P.5.77: In Exercises 7782, factor the expression by removing thecommon fact...
 P.5.78: In Exercises 7782, factor the expression by removing thecommon fact...
 P.5.79: In Exercises 7782, factor the expression by removing thecommon fact...
 P.5.80: In Exercises 7782, factor the expression by removing thecommon fact...
 P.5.81: In Exercises 7782, factor the expression by removing thecommon fact...
 P.5.82: In Exercises 7782, factor the expression by removing thecommon fact...
 P.5.83: In Exercises 83 and 84, simplify the expression3x13 x233x23
 P.5.84: In Exercises 83 and 84, simplify the expressionx 31 x 212 2x1 x 212x4
 P.5.85: In Exercises 85 88, simplify the difference quotient. 1x h 1xh
 P.5.86: In Exercises 85 88, simplify the difference quotient. 1x h2 1x 2h
 P.5.87: In Exercises 85 88, simplify the difference quotient. 1x h 4 1x 4h
 P.5.88: In Exercises 85 88, simplify the difference quotient. x hx h 1 xx 1h
 P.5.89: In Exercises 8994, simplify the difference quotient byrationalizing...
 P.5.90: In Exercises 8994, simplify the difference quotient byrationalizing...
 P.5.91: In Exercises 8994, simplify the difference quotient byrationalizing...
 P.5.92: In Exercises 8994, simplify the difference quotient byrationalizing...
 P.5.93: In Exercises 8994, simplify the difference quotient byrationalizing...
 P.5.94: In Exercises 8994, simplify the difference quotient byrationalizing...
 P.5.95: PROBABILITY In Exercises 95 and 96, consider an experimentin which ...
 P.5.96: PROBABILITY In Exercises 95 and 96, consider an experimentin which ...
 P.5.97: RATE A digital copier copies in color at a rate of50 pages per minu...
 P.5.98: RATE After working together for t hours on acommon task, two worker...
 P.5.99: FINANCE In Exercises 99 and 100, the formula thatapproximates the a...
 P.5.100: FINANCE In Exercises 99 and 100, the formula thatapproximates the a...
 P.5.101: REFRIGERATION When food (at room temperature)is placed in a refrige...
 P.5.102: INTERACTIVE MONEY MANAGEMENT The tableshows the projected numbers o...
 P.5.103: TRUE OR FALSE? In Exercises 103 and 104, determinewhether the state...
 P.5.104: TRUE OR FALSE? In Exercises 103 and 104, determinewhether the state...
 P.5.105: THINK ABOUT IT How do you determine whether arational expression is...
 P.5.106: CAPSTONE In your own words, explain how todivide rational expressions.
Solutions for Chapter P.5: Rational Expressions
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter P.5: Rational Expressions
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9781439048696. Chapter P.5: Rational Expressions includes 106 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 106 problems in chapter P.5: Rational Expressions have been answered, more than 33644 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 8.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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 •

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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