 5.7.1: Factor. See Examples 1 and 2.x2 + 6x + 9
 5.7.2: Factor. See Examples 1 and 2.x2  10x + 25
 5.7.3: Factor. See Examples 1 and 2.4x2  12x + 9
 5.7.4: Factor. See Examples 1 and 2.9a2  30a + 25
 5.7.5: Factor. See Examples 1 and 2.25x2 + 10x + 1
 5.7.6: Factor. See Examples 1 and 2.4a2 + 12a + 9
 5.7.7: Factor. See Examples 1 and 2.3x2  24x + 48
 5.7.8: Factor. See Examples 1 and 2.2x2 + 28x + 98
 5.7.9: Factor. See Examples 1 and 2.9y2x2 + 12yx2 + 4x2
 5.7.10: Factor. See Examples 1 and 2.4x2y3  4xy3 + y3
 5.7.11: Factor. See Examples 1 and 2.16x2  56xy + 49y2
 5.7.12: Factor. See Examples 1 and 2.81x2 + 36xy + 4y2
 5.7.13: Factor. See Examples 3 through 5x2  25
 5.7.14: Factor. See Examples 3 through 5y2  100
 5.7.15: Factor. See Examples 3 through 519  4z2
 5.7.16: Factor. See Examples 3 through 5116  y2
 5.7.17: Factor. See Examples 3 through 51y + 222  49
 5.7.18: Factor. See Examples 3 through 51x  122  z2
 5.7.19: Factor. See Examples 3 through 564x2  100
 5.7.20: Factor. See Examples 3 through 54x2  36
 5.7.21: Factor. See Examples 3 through 51x + 2y22  9
 5.7.22: Factor. See Examples 3 through 513x + y22  25
 5.7.23: Factor. See Examples 3 through 5x2 + 6x + 9  y2
 5.7.24: Factor. See Examples 3 through 5x2 + 12x + 36  y2
 5.7.25: Factor. See Examples 3 through 5x2 + 16x + 64  x4
 5.7.26: Factor. See Examples 3 through 5x2 + 20x + 100  x4
 5.7.27: Factor. See Examples 6 through 9.x3 + 27
 5.7.28: Factor. See Examples 6 through 9.y3 + 1
 5.7.29: Factor. See Examples 6 through 9.z3  1
 5.7.30: Factor. See Examples 6 through 9.x3  8
 5.7.31: Factor. See Examples 6 through 9.m3 + n3
 5.7.32: Factor. See Examples 6 through 9.p3 + 125q3
 5.7.33: Factor. See Examples 6 through 9.27y2  x3y2
 5.7.34: Factor. See Examples 6 through 9.64q2  q2p3
 5.7.35: Factor. See Examples 6 through 9.8ab3 + 27a4
 5.7.36: Factor. See Examples 6 through 9.a3b + 8b4
 5.7.37: Factor. See Examples 6 through 9.250y3  16x3
 5.7.38: Factor. See Examples 6 through 9.54y3  128
 5.7.39: Factor completely. See Examples 1 through 9.x2  12x + 36
 5.7.40: Factor completely. See Examples 1 through 9.x2  18x + 81
 5.7.41: Factor completely. See Examples 1 through 9.18x2y  2y
 5.7.42: Factor completely. See Examples 1 through 9.12xy2  108x
 5.7.43: Factor completely. See Examples 1 through 9.9x2  49
 5.7.44: Factor completely. See Examples 1 through 9.25x2  4
 5.7.45: Factor completely. See Examples 1 through 9.x4  1
 5.7.46: Factor completely. See Examples 1 through 9.x4  256
 5.7.47: Factor completely. See Examples 1 through 9.x6  y3
 5.7.48: Factor completely. See Examples 1 through 9.x3  y6
 5.7.49: Factor completely. See Examples 1 through 9.8x3 + 27y3
 5.7.50: Factor completely. See Examples 1 through 9.125x3 + 8y3
 5.7.51: Factor completely. See Examples 1 through 9.4x2 + 4x + 1  z2
 5.7.52: Factor completely. See Examples 1 through 9.9y2 + 12y + 4  x2
 5.7.53: Factor completely. See Examples 1 through 9.3x6y2 + 81y2
 5.7.54: Factor completely. See Examples 1 through 9.x2y9 + x2y3
 5.7.55: Factor completely. See Examples 1 through 9.n3  127
 5.7.56: Factor completely. See Examples 1 through 9.p3 +1125
 5.7.57: Factor completely. See Examples 1 through 9.16y2 + 64
 5.7.58: Factor completely. See Examples 1 through 9.12y2 + 108
 5.7.59: Factor completely. See Examples 1 through 9.x2  10x + 25  y2
 5.7.60: Factor completely. See Examples 1 through 9.x2  18x + 81  y2
 5.7.61: Factor completely. See Examples 1 through 9.a3b3 + 125
 5.7.62: Factor completely. See Examples 1 through 9.x3y3 + 216
 5.7.63: Factor completely. See Examples 1 through 9.x225  y29
 5.7.64: Factor completely. See Examples 1 through 9.a24  b249
 5.7.65: Factor completely. See Examples 1 through 9.1x + y23 + 125
 5.7.66: Factor completely. See Examples 1 through 9.1r + s23 + 27
 5.7.67: Solve the following equations. See Section 2.1.x  5 = 0
 5.7.68: Solve the following equations. See Section 2.1.x + 7 = 0
 5.7.69: Solve the following equations. See Section 2.1.3x + 1 = 0
 5.7.70: Solve the following equations. See Section 2.1.5x  15 = 0
 5.7.71: Solve the following equations. See Section 2.1.2x = 0
 5.7.72: Solve the following equations. See Section 2.1.3x = 0
 5.7.73: Solve the following equations. See Section 2.1.5x + 25 = 0
 5.7.74: Solve the following equations. See Section 2.1.4x  16 = 0
 5.7.75: Determine whether each polynomial is factored completely. See the C...
 5.7.76: Determine whether each polynomial is factored completely. See the C...
 5.7.77: Determine whether each polynomial is factored completely. See the C...
 5.7.78: Determine whether each polynomial is factored completely. See the C...
 5.7.79: A manufacturer of metal washers needs to determine the crosssectio...
 5.7.80: Express the area of the shaded region as a polynomial. Factor the p...
 5.7.81: Express the volume of each solid as a polynomial. To do so, subtrac...
 5.7.82: Express the volume of each solid as a polynomial. To do so, subtrac...
 5.7.83: Find the value of c that makes each trinomial a perfect square trin...
 5.7.84: Find the value of c that makes each trinomial a perfect square trin...
 5.7.85: Find the value of c that makes each trinomial a perfect square trin...
 5.7.86: Find the value of c that makes each trinomial a perfect square trin...
 5.7.87: Find the value of c that makes each trinomial a perfect square trin...
 5.7.88: Find the value of c that makes each trinomial a perfect square trin...
 5.7.89: Factor x6  1 completely, using the following methods fromthis chap...
 5.7.90: Factor x12  1 completely, using the following methods fromthis cha...
 5.7.91: Factor. Assume that variables used as exponents represent positive ...
 5.7.92: Factor. Assume that variables used as exponents represent positive ...
 5.7.93: Factor. Assume that variables used as exponents represent positive ...
 5.7.94: Factor. Assume that variables used as exponents represent positive ...
 5.7.95: Factor. Assume that variables used as exponents represent positive ...
 5.7.96: Factor. Assume that variables used as exponents represent positive ...
Solutions for Chapter 5.7: Factoring by Special Products
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 5.7: Factoring by Special Products
Get Full SolutionsIntermediate Algebra was written by and is associated to the ISBN: 9780321785046. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.7: Factoring by Special Products includes 96 full stepbystep solutions. Since 96 problems in chapter 5.7: Factoring by Special Products have been answered, more than 59802 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

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).

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).

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

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.

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