 6.2.1: P R + Q R = ______________: To add rational expressions with the sa...
 6.2.2: P R  Q R = ______________: To subtract rational expressions with t...
 6.2.3: x 3  5  y 3 = ______________
 6.2.4: When adding or subtracting rational expressions with denominators t...
 6.2.5: The first step in finding the least common denominator of 7 5x2 + 1...
 6.2.6: Consider the following subtraction problem: x  1 x2 + x  6  x  ...
 6.2.7: An equivalent expression for 7 15x with a denominator of 30x2 can b...
 6.2.8: An equivalent expression for 3y + 2 y  5 with a denominator of (3y...
 6.2.9: An equivalent expression for x  2y 3y  x with a denominator of x ...
 6.2.10: 3x 7x  4  2x  1 7x  4
 6.2.11: x2  2 x2 + 6x  7  19  4x x2 + 6x  7
 6.2.12: x2 + 6x + 2 x2 + x  6  2x  1 x2 + x  6
 6.2.13: 20y2 + 5y + 1 6y2 + y  2  8y2  12y  5 6y2 + y  2
 6.2.14: y2 + 3y  6 y2  5y + 4  4y  4  2y2 y2  5y + 4
 6.2.15: 2x3  3y3 x2  y2  x3  2y3 x2  y2
 6.2.16: 4y3  3x3 y2  x2  3y3  2x3 y2  x2
 6.2.17: 11 25x2 and 14 35x
 6.2.18: 7 15x2 and 9 24x
 6.2.19: 2 x  5 and 3 x2  25
 6.2.20: 2 x + 3 and 5 x2  9
 6.2.21: 7 y2  100 and 13 y(y  10)
 6.2.22: 7 y2  4 and 15 y(y + 2)
 6.2.23: 8 x2  16 and x x2  8x + 16
 6.2.24: 3 x2  25 and x x2  10x + 25
 6.2.25: 7 y2  5y  6 and y y2  4y  5
 6.2.26: 3 y2  y  20 and y 2y2 + 7y  4
 6.2.27: 7y 2y2 + 7y + 6 , 3 y2  4 , and 7y 2y2  3y  2
 6.2.28: 5y y2  9 , 8 y2 + 6y + 9 , and 5y 2y2 + 5y  3
 6.2.29: 3 5x2 + 10 x
 6.2.30: 7 2x2 + 4 x
 6.2.31: 4 x  2 + 3 x + 1
 6.2.32: 2 x  3 + 7 x + 2
 6.2.33: 3x x2 + x  2 + 2 x2  4x + 3
 6.2.34: 7x x2 + 2x  8 + 3 x2  3x + 2
 6.2.35: x  6 x + 5 + x + 5 x  6
 6.2.36: x  2 x + 7 + x + 7 x  2
 6.2.37: 3x x2  25  4 x + 5
 6.2.38: 8x x2  16  5 x + 4
 6.2.39: 3y + 7 y2  5y + 6  3 y  3
 6.2.40: 2y + 9 y2  7y + 12  2 y  3
 6.2.41: x2  6 x2 + 9x + 18  x  4 x + 6
 6.2.42: x2  39 x2 + 3x  10  x  7 x  2
 6.2.43: 4x + 1 x2 + 7x + 12 + 2x + 3 x2 + 5x + 4
 6.2.44: 3x  2 x2  x  6 + 4x  3 x2  9
 6.2.45: x + 4 x2  x  2  2x + 3 x2 + 2x  8
 6.2.46: 2x + 1 x2  7x + 6  x + 3 x2  5x  6
 6.2.47: 4 + 1 x  3
 6.2.48: 7 + 1 x  5
 6.2.49: y  7 y2  16 + 7  y 16  y2
 6.2.50: y  3 y2  25 + y  3 25  y2
 6.2.51: x + 7 3x + 6 + x 4  x2
 6.2.52: x + 5 4x + 12 + x 9  x2
 6.2.53: 2x x  4 + 64 x2  16  2x x + 4
 6.2.54: x x  3 + x + 2 x2  2x  3  4 x + 1
 6.2.55: 5x x2  y2  7 y  x
 6.2.56: 9x x2  y2  10 y  x
 6.2.57: 3 5x + 6  4 x  2 + x2  x 5x2  4x  12
 6.2.58: x  1 x2 + 2x + 1  3 2x  2 + x x2  1
 6.2.59: 3x  y x2  9xy + 20y2 + 2y x2  25y2
 6.2.60: x + 2y x2 + 4xy + 4y2  2x x2  4y2
 6.2.61: 3x x2  4 + 5x x2 + x  2  3 x2  4x + 4
 6.2.62: 1 x + 4 x2  4  2 x2  2x
 6.2.63: 6a + 5b 6a2 + 5ab  4b2  a + 2b 9a2  16b2
 6.2.64: 5a  b a2 + ab  2b2  3a + 2b a2 + 5ab  6b2
 6.2.65: 1 m2 + m  2  3 2m2 + 3m  2 + 2 2m2  3m + 1
 6.2.66: 5 2m2  5m  3 + 3 2m2 + 5m + 2  1 m2  m  6
 6.2.67: 2x + 3 x + 1 # x2 + 4x  5 2x2 + x  3  2 x + 2
 6.2.68: 1 x2  2x  8 , a 1 x  4  1 x + 2 b
 6.2.69: a2  6 x + 1 b a1 + 3 x  2 b
 6.2.70: a4  3 x + 2 b a1 + 5 x  1 b
 6.2.71: a 1 x + h  1 x b , h
 6.2.72: a 5 x  5  2 x + 3 b , (3x + 25)
 6.2.73: 1 a3  b3 # ac + ad  bc  bd 1  c  d a2 + ab + b2
 6.2.74: ab a2 + ab + b2 + ac  ad  bc + bd ac  ad + bc  bd , a3  b3 a3 ...
 6.2.75: If f (x) = 2x  3 x + 5 and g(x) = x2  4x  19 x2 + 8x + 15 , find...
 6.2.76: If f(x) = 2x  1 x2 + x  6 and g(x) = x + 2 x2 + 5x + 6 , find (f ...
 6.2.77: Find and interpret T(0). Round to the nearest hour. Identify your s...
 6.2.78: Find and interpret T(5). Round to the nearest hour. Identify your s...
 6.2.79: Find a simplified form of T(x) by adding the rational expressions i...
 6.2.80: Find a simplified form of T(x) by adding the rational expressions i...
 6.2.81: Use the graph to answer this question. If you want the driving time...
 6.2.82: Use the graph to answer this question. If you want the driving time...
 6.2.83: In the section opener, we saw that the rational function f(x) = 27,...
 6.2.84: In Exercises 8485, express the perimeter of each rectangle as a sin...
 6.2.85: In Exercises 8485, express the perimeter of each rectangle as a sin...
 6.2.86: Explain how to add rational expressions when denominators are the s...
 6.2.87: Explain how to subtract rational expressions when denominators are ...
 6.2.88: Explain how to find the least common denominator for denominators o...
 6.2.89: Explain how to add rational expressions that have different denomin...
 6.2.90: Explain how to add rational expressions when denominators are oppos...
 6.2.91: 1 a + 1 b = 1 a + b
 6.2.92: 1 x + 3 7 = 4 x + 7
 6.2.93: When a numerator is being subtracted, I find that inserting parenth...
 6.2.94: The reason I can rewrite rational expressions with a common denomin...
 6.2.95: The fastest way for me to add 5 x  7 + 3 7  x is by using (x  7)...
 6.2.96: Although 2x3 + 11x2 x + 3 + 5x3 + 4x2 x + 3 looks more complicated ...
 6.2.97: 2 x + 3 + 3 x + 4 = 5 2x + 7
 6.2.98: a b + a c = a b + c
 6.2.99: 6 + 1 x = 7 x
 6.2.100: 1 x + 3 + x + 3 2 = 1 (x + 3) + (x + 3) 2 = 1 + 1 2 = 3 2
 6.2.101: 1 xn  1  1 xn + 1  1 x2n  1
 6.2.102: a1  1 x b a1  1 x + 1 b a1  1 x + 2 b a1  1 x + 3 b
 6.2.103: (x  y)1 + (x  y)2
 6.2.104: Simplify: 3x2y 2 y3 2 . (Section 1.6, Example 9)
 6.2.105: Solve: 3x  1 14. (Section 4.3, Example 4)
 6.2.106: Factor completely: 50x3  18x. (Section 5.5, Example 2)
 6.2.107: Multiply and simplify: x2y2 1 x + y x2
 6.2.108: Multiply and simplify: x(x + h)a 1 x + h  1 x b.
 6.2.109: Divide: x2  1 x2 , x2  4x + 3 x2 .
Solutions for Chapter 6.2: Adding and Subtracting Rational Expressions
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6.2: Adding and Subtracting Rational Expressions
Get Full SolutionsSince 109 problems in chapter 6.2: Adding and Subtracting Rational Expressions have been answered, more than 12818 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Chapter 6.2: Adding and Subtracting Rational Expressions includes 109 full stepbystep solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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.

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).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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