 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 9115 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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

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).

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

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 •

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.
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