 6.1.1: Find the greatest common factor for each list of numbers. See Examp...
 6.1.2: Find the greatest common factor for each list of numbers. See Examp...
 6.1.3: Find the greatest common factor for each list of numbers. See Examp...
 6.1.4: Find the greatest common factor for each list of numbers. See Examp...
 6.1.5: Find the greatest common factor for each list of numbers. See Examp...
 6.1.6: Find the greatest common factor for each list of numbers. See Examp...
 6.1.7: Find the greatest common factor for each list of terms. See Example...
 6.1.8: Find the greatest common factor for each list of terms. See Example...
 6.1.9: Find the greatest common factor for each list of terms. See Example...
 6.1.10: Find the greatest common factor for each list of terms. See Example...
 6.1.11: Find the greatest common factor for each list of terms. See Example...
 6.1.12: Find the greatest common factor for each list of terms. See Example...
 6.1.13: Find the greatest common factor for each list of terms. See Example...
 6.1.14: Find the greatest common factor for each list of terms. See Example...
 6.1.15: An expression is factored when it is written as a product, not a su...
 6.1.16: An expression is factored when it is written as a product, not a su...
 6.1.17: An expression is factored when it is written as a product, not a su...
 6.1.18: An expression is factored when it is written as a product, not a su...
 6.1.19: Complete each factoring by writing each polynomial as the product o...
 6.1.20: Complete each factoring by writing each polynomial as the product o...
 6.1.21: Complete each factoring by writing each polynomial as the product o...
 6.1.22: Complete each factoring by writing each polynomial as the product o...
 6.1.23: Complete each factoring by writing each polynomial as the product o...
 6.1.24: Complete each factoring by writing each polynomial as the product o...
 6.1.25: Complete each factoring by writing each polynomial as the product o...
 6.1.26: Complete each factoring by writing each polynomial as the product o...
 6.1.27: Complete each factoring by writing each polynomial as the product o...
 6.1.28: Complete each factoring by writing each polynomial as the product o...
 6.1.29: Complete each factoring by writing each polynomial as the product o...
 6.1.30: Complete each factoring by writing each polynomial as the product o...
 6.1.31: How can you check your answer when you factor a polynomial?
 6.1.32: A student factored as WHAT WENT WRONG? Factor correctly.
 6.1.33: Write in factored form by factoring out the greatest common factor....
 6.1.34: Write in factored form by factoring out the greatest common factor....
 6.1.35: Write in factored form by factoring out the greatest common factor....
 6.1.36: Write in factored form by factoring out the greatest common factor....
 6.1.37: Write in factored form by factoring out the greatest common factor....
 6.1.38: Write in factored form by factoring out the greatest common factor....
 6.1.39: Write in factored form by factoring out the greatest common factor....
 6.1.40: Write in factored form by factoring out the greatest common factor....
 6.1.41: Write in factored form by factoring out the greatest common factor....
 6.1.42: Write in factored form by factoring out the greatest common factor....
 6.1.43: Write in factored form by factoring out the greatest common factor....
 6.1.44: Write in factored form by factoring out the greatest common factor....
 6.1.45: Write in factored form by factoring out the greatest common factor....
 6.1.46: Write in factored form by factoring out the greatest common factor....
 6.1.47: Write in factored form by factoring out the greatest common factor....
 6.1.48: Write in factored form by factoring out the greatest common factor....
 6.1.49: Write in factored form by factoring out the greatest common factor....
 6.1.50: Write in factored form by factoring out the greatest common factor....
 6.1.51: Write in factored form by factoring out the greatest common factor....
 6.1.52: Write in factored form by factoring out the greatest common factor....
 6.1.53: Write in factored form by factoring out the greatest common factor....
 6.1.54: Write in factored form by factoring out the greatest common factor....
 6.1.55: Write in factored form by factoring out the greatest common factor....
 6.1.56: Write in factored form by factoring out the greatest common factor....
 6.1.57: Write in factored form by factoring out the greatest common factor....
 6.1.58: Write in factored form by factoring out the greatest common factor....
 6.1.59: Write in factored form by factoring out the greatest common factor....
 6.1.60: Write in factored form by factoring out the greatest common factor....
 6.1.61: Determine whether each expression is in factored form or is not in ...
 6.1.62: Determine whether each expression is in factored form or is not in ...
 6.1.63: Determine whether each expression is in factored form or is not in ...
 6.1.64: Determine whether each expression is in factored form or is not in ...
 6.1.65: Determine whether each expression is in factored form or is not in ...
 6.1.66: Determine whether each expression is in factored form or is not in ...
 6.1.67: Why is it not possible to factor the expression in Exercise 65?
 6.1.68: A student factored x3 + 4x2  2x  8 as follows= x21x + 42 + 21x ...
 6.1.69: Factor by grouping. See Examples 5 and 6.p2 + 4p + pq + 4q
 6.1.70: Factor by grouping. See Examples 5 and 6.m2 + 2m + mn + 2n
 6.1.71: Factor by grouping. See Examples 5 and 6.a2  2a + ab  2b
 6.1.72: Factor by grouping. See Examples 5 and 6.y2 6y + yw  6w
 6.1.73: Factor by grouping. See Examples 5 and 6.7z 2 + 14z  az  2a
 6.1.74: Factor by grouping. See Examples 5 and 6.5m2 7z + 15mp  2mr  6pr
 6.1.75: Factor by grouping. See Examples 5 and 6.18r2 + 12ry  3xr  2xy
 6.1.76: Factor by grouping. See Examples 5 and 6.8s2  4st + 6sy  3yt
 6.1.77: Factor by grouping. See Examples 5 and 6.3a3 + 3ab2 + 2a2b + 2b3
 6.1.78: Factor by grouping. See Examples 5 and 6.4x3 + 3x2y + 4xy2 + 3y3
 6.1.79: Factor by grouping. See Examples 5 and 6.12  4a  3b + ab
 6.1.80: Factor by grouping. See Examples 5 and 6.6  3x  2y + xy
 6.1.81: Factor by grouping. See Examples 5 and 6.16m3  4m2p2  4mp + p3
 6.1.82: Factor by grouping. See Examples 5 and 6.10t3  2t2s2  5ts + s3
 6.1.83: Factor by grouping. See Examples 5 and 6.y22 + 3x + 3y + xyw
 6.1.84: Factor by grouping. See Examples 5 and 6.m2 y + 14p + 7m + 2mp
 6.1.85: Factor by grouping. See Examples 5 and 6.5m  6p  2mp + 15
 6.1.86: Factor by grouping. See Examples 5 and 6.7y  9x  3xy + 21
 6.1.87: Factor by grouping. See Examples 5 and 6.18r2 2ty + 12ry  3rt
 6.1.88: Factor by grouping. See Examples 5 and 6.12a2  4bc + 16ac  3ab
 6.1.89: Factor by grouping. See Examples 5 and 6.a  5  3 + 2a5b  6b
 6.1.90: Factor by grouping. See Examples 5 and 6.b3  2 + 5ab3  10a
 6.1.91: In many cases, the choice of which pairs of terms to group when fac...
 6.1.92: In many cases, the choice of which pairs of terms to group when fac...
 6.1.93: In many cases, the choice of which pairs of terms to group when fac...
 6.1.94: In many cases, the choice of which pairs of terms to group when fac...
 6.1.95: Find each product. See Section 5.5.1x + 621x  92
 6.1.96: Find each product. See Section 5.5.1x  321x  62
 6.1.97: Find each product. See Section 5.5.1x + 221x + 7
 6.1.98: Find each product. See Section 5.5.2x1x + 521x  1
 6.1.99: Find each product. See Section 5.5.2x21x2 + 3x + 52
 6.1.100: Find each product. See Section 5.5.5x212x2  4x  92
Solutions for Chapter 6.1: The Greatest Common Factor; Factoring by Grouping
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 6.1: The Greatest Common Factor; Factoring by Grouping
Get Full SolutionsChapter 6.1: The Greatest Common Factor; Factoring by Grouping includes 100 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 100 problems in chapter 6.1: The Greatest Common Factor; Factoring by Grouping have been answered, more than 38056 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. Beginning Algebra was written by and is associated to the ISBN: 9780321673480.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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 •

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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 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.

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

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.