 7.2.1: Match each multiplication problem in Column I with the correct prod...
 7.2.2: Match each division problem in Column I with the correct quotient i...
 7.2.3: Multiply. Write each answer in lowest terms. See Examples 1 and 2.1...
 7.2.4: Multiply. Write each answer in lowest terms. See Examples 1 and 2.2...
 7.2.5: Multiply. Write each answer in lowest terms. See Examples 1 and 2.1...
 7.2.6: Multiply. Write each answer in lowest terms. See Examples 1 and 2.1...
 7.2.7: Multiply. Write each answer in lowest terms. See Examples 1 and 2.2...
 7.2.8: Multiply. Write each answer in lowest terms. See Examples 1 and 2.4...
 7.2.9: Multiply. Write each answer in lowest terms. See Examples 1 and 2.1...
 7.2.10: Multiply. Write each answer in lowest terms. See Examples 1 and 2.1...
 7.2.11: Multiply. Write each answer in lowest terms. See Examples 1 and 2.t...
 7.2.12: Multiply. Write each answer in lowest terms. See Examples 1 and 2.z...
 7.2.13: Multiply. Write each answer in lowest terms. See Examples 1 and 2.3...
 7.2.14: Multiply. Write each answer in lowest terms. See Examples 1 and 2.1...
 7.2.15: Divide. Write each answer in lowest terms. See Examples 4 and 5.9z4...
 7.2.16: Divide. Write each answer in lowest terms. See Examples 4 and 5.35x...
 7.2.17: Divide. Write each answer in lowest terms. See Examples 4 and 5.4t ...
 7.2.18: Divide. Write each answer in lowest terms. See Examples 4 and 5.12...
 7.2.19: Divide. Write each answer in lowest terms. See Examples 4 and 5.32y...
 7.2.20: Divide. Write each answer in lowest terms. See Examples 4 and 5.4m ...
 7.2.21: Divide. Write each answer in lowest terms. See Examples 4 and 5.7t ...
 7.2.22: Divide. Write each answer in lowest terms. See Examples 4 and 5.8z ...
 7.2.23: Divide. Write each answer in lowest terms. See Examples 4 and 5.2xx...
 7.2.24: Divide. Write each answer in lowest terms. See Examples 4 and 5.y2y...
 7.2.25: Divide. Write each answer in lowest terms. See Examples 4 and 5.1x ...
 7.2.26: Divide. Write each answer in lowest terms. See Examples 4 and 5.2aa...
 7.2.27: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.28: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.29: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.30: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.31: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.32: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.33: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.34: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.35: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.36: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.37: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.38: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.39: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.40: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.41: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.42: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.43: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.44: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.45: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.46: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.47: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.48: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.49: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.50: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.51: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.52: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.53: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.54: Multiply or divide. Write each answer in lowest terms. See Examples...
 7.2.55: Exercises 5560 involve grouping symbols (Section 1.2), factoring by...
 7.2.56: Exercises 5560 involve grouping symbols (Section 1.2), factoring by...
 7.2.57: Exercises 5560 involve grouping symbols (Section 1.2), factoring by...
 7.2.58: Exercises 5560 involve grouping symbols (Section 1.2), factoring by...
 7.2.59: Exercises 5560 involve grouping symbols (Section 1.2), factoring by...
 7.2.60: Exercises 5560 involve grouping symbols (Section 1.2), factoring by...
 7.2.61: If the rational expression 5x2y 32pq represents the area of a recta...
 7.2.62: k If you are given the following problem, what must be the polynomi...
 7.2.63: Write the prime factored form of each number. See Section 1.1.18
 7.2.64: Write the prime factored form of each number. See Section 1.1.48
 7.2.65: Write the prime factored form of each number. See Section 1.1.108
 7.2.66: Write the prime factored form of each number. See Section 1.1.60
 7.2.67: Find the greatest common factor of each group of terms. See Section...
 7.2.68: Find the greatest common factor of each group of terms. See Section...
 7.2.69: Find the greatest common factor of each group of terms. See Section...
 7.2.70: Find the greatest common factor of each group of terms. See Section...
Solutions for Chapter 7.2: Multiplying and Dividing Rational Expressions
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 7.2: Multiplying and Dividing Rational Expressions
Get Full SolutionsSince 70 problems in chapter 7.2: Multiplying and Dividing Rational Expressions have been answered, more than 37928 students have viewed full stepbystep solutions from this chapter. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.2: Multiplying and Dividing Rational Expressions includes 70 full stepbystep solutions.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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 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!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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).

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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.

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.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.