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 7.3.7.1.313: In Exercises 7576, find the perimeter of each rectangle. 7 x 4 inch...
 7.3.7.1.314: Explain how to add rational expressions when denominators are the s...
 7.3.7.1.315: Explain how to subtract rational expressions when denominators are ...
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 7.3.7.1.328: In Exercises 9195, find the missing expression. 2x x 3 x 3 4x 1 x 3
 7.3.7.1.329: In Exercises 9195, find the missing expression. 3x x 2 x 2 6 17x x 2
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 7.3.7.1.336: Subtract: 13 15 8 45 . (Section 1.2, Example 9)
 7.3.7.1.337: Factor completely: 81x4 1. (Section 6.4, Example 4)
 7.3.7.1.338: Divide: 3x3 2x2 26x 15 x 3 . (Section 5.6, Example 2)
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Solutions for Chapter 7.3: Adding and Subtracting RationalExpressions with the SameDenominator
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 7.3: Adding and Subtracting RationalExpressions with the SameDenominator
Get Full SolutionsThis textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Chapter 7.3: Adding and Subtracting RationalExpressions with the SameDenominator includes 110 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 110 problems in chapter 7.3: Adding and Subtracting RationalExpressions with the SameDenominator have been answered, more than 71142 students have viewed full stepbystep solutions from this chapter. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941.

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

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.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.