 6.4.6.1.374: Fill in each blank so that the resulting statement is true. The for...
 6.4.6.1.375: Fill in each blank so that the resulting statement is true. A formu...
 6.4.6.1.376: Fill in each blank so that the resulting statement is true. A formu...
 6.4.6.1.377: Fill in each blank so that the resulting statement is true. The for...
 6.4.6.1.378: Fill in each blank so that the resulting statement is true. The for...
 6.4.6.1.379: Fill in each blank so that the resulting statement is true. 36x2 49...
 6.4.6.1.380: Fill in each blank so that the resulting statement is true. x2 12x ...
 6.4.6.1.381: Fill in each blank so that the resulting statement is true. 16x2 24...
 6.4.6.1.382: Fill in each blank so that the resulting statement is true. x3 8 (x...
 6.4.6.1.383: Fill in each blank so that the resulting statement is true. x3 27 (...
 6.4.6.1.384: Fill in each blank so that the resulting statement is true. True or...
 6.4.6.1.385: Fill in each blank so that the resulting statement is true. True or...
 6.4.6.1.386: Fill in each blank so that the resulting statement is true. True or...
 6.4.6.1.387: Fill in each blank so that the resulting statement is true. True or...
 6.4.6.1.388: Fill in each blank so that the resulting statement is true. True or...
 6.4.6.1.389: In Exercises 126, factor each difference of two squares. x2 25
 6.4.6.1.390: In Exercises 126, factor each difference of two squares. x2 16
 6.4.6.1.391: In Exercises 126, factor each difference of two squares. y2 1
 6.4.6.1.392: In Exercises 126, factor each difference of two squares. y2 9
 6.4.6.1.393: In Exercises 126, factor each difference of two squares. 4x2 9
 6.4.6.1.394: In Exercises 126, factor each difference of two squares. 9x2 25
 6.4.6.1.395: In Exercises 126, factor each difference of two squares. 25 x2
 6.4.6.1.396: In Exercises 126, factor each difference of two squares. 16 x2
 6.4.6.1.397: In Exercises 126, factor each difference of two squares. 1 49x2
 6.4.6.1.398: In Exercises 126, factor each difference of two squares. 1 64x2
 6.4.6.1.399: In Exercises 126, factor each difference of two squares. 9 25y2
 6.4.6.1.400: In Exercises 126, factor each difference of two squares. 16 49y2
 6.4.6.1.401: In Exercises 126, factor each difference of two squares. x4 9
 6.4.6.1.402: In Exercises 126, factor each difference of two squares. x4 25
 6.4.6.1.403: In Exercises 126, factor each difference of two squares. 49y4 16
 6.4.6.1.404: In Exercises 126, factor each difference of two squares. 49y4 25
 6.4.6.1.405: In Exercises 126, factor each difference of two squares. x10 9
 6.4.6.1.406: In Exercises 126, factor each difference of two squares. x10 1
 6.4.6.1.407: In Exercises 126, factor each difference of two squares. 25x2 16y2
 6.4.6.1.408: In Exercises 126, factor each difference of two squares. 9x2 25y2
 6.4.6.1.409: In Exercises 126, factor each difference of two squares. x4 y10
 6.4.6.1.410: In Exercises 126, factor each difference of two squares. x14 y4
 6.4.6.1.411: In Exercises 126, factor each difference of two squares. x4 16
 6.4.6.1.412: In Exercises 126, factor each difference of two squares. x4 1
 6.4.6.1.413: In Exercises 126, factor each difference of two squares. 16x4 81
 6.4.6.1.414: In Exercises 126, factor each difference of two squares. 81x4 1
 6.4.6.1.415: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.416: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.417: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.418: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.419: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.420: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.421: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.422: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.423: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.424: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.425: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.426: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.427: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.428: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.429: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.430: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.431: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.432: In Exercises 2744, factor completely, or state that the polynomial ...
 6.4.6.1.433: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.434: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.435: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.436: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.437: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.438: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.439: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.440: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.441: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.442: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.443: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.444: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.445: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.446: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.447: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.448: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.449: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.450: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.451: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.452: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.453: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.454: In Exercises 4566, factor any perfect square trinomials, or state t...
 6.4.6.1.455: In Exercises 6778, factor completely. 12x2 12x 3
 6.4.6.1.456: In Exercises 6778, factor completely. 18x2 24x 8
 6.4.6.1.457: In Exercises 6778, factor completely. 9x3 6x2 x
 6.4.6.1.458: In Exercises 6778, factor completely. 25x3 10x2 x
 6.4.6.1.459: In Exercises 6778, factor completely. 2y2 4y 2
 6.4.6.1.460: In Exercises 6778, factor completely. 2y2 40y 200
 6.4.6.1.461: In Exercises 6778, factor completely. 2y3 28y2 98y
 6.4.6.1.462: In Exercises 6778, factor completely. 50y3 20y2 2y
 6.4.6.1.463: In Exercises 6778, factor completely. 6x2 24x 24
 6.4.6.1.464: In Exercises 6778, factor completely. 5x2 30x 45
 6.4.6.1.465: In Exercises 6778, factor completely. 16y3 16y2 4y
 6.4.6.1.466: In Exercises 6778, factor completely. 45y3 30y2 5y
 6.4.6.1.467: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.468: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.469: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.470: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.471: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.472: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.473: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.474: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.475: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.476: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.477: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.478: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.479: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.480: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.481: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.482: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.483: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.484: In Exercises 7996, factor using the formula for the sum or differen...
 6.4.6.1.485: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.486: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.487: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.488: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.489: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.490: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.491: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.492: In Exercises 97104, factor completely. (Hint on Exercises 97102: Fa...
 6.4.6.1.493: Divide x3 x2 5x 3 by x 3. Use the quotient to factor x3 x2 5x 3 com...
 6.4.6.1.494: Divide x3 4x2 3x 18 by x 2. Use the quotient to factor x3 4x2 3x 18...
 6.4.6.1.495: In Exercises 107110, find the formula for the area of the shaded bl...
 6.4.6.1.496: In Exercises 107110, find the formula for the area of the shaded bl...
 6.4.6.1.497: In Exercises 107110, find the formula for the area of the shaded bl...
 6.4.6.1.498: In Exercises 107110, find the formula for the area of the shaded bl...
 6.4.6.1.499: Explain how to factor the difference of two squares. Provide an exa...
 6.4.6.1.500: What is a perfect square trinomial and how is it factored?
 6.4.6.1.501: Explain why x2 1 is factorable, but x2 1 is not.
 6.4.6.1.502: Explain how to factor x3 1.
 6.4.6.1.503: In Exercises 115118, determine whether each statement makes sense o...
 6.4.6.1.504: In Exercises 115118, determine whether each statement makes sense o...
 6.4.6.1.505: In Exercises 115118, determine whether each statement makes sense o...
 6.4.6.1.506: In Exercises 115118, determine whether each statement makes sense o...
 6.4.6.1.507: In Exercises 119122, determine whether each statement is true or fa...
 6.4.6.1.508: In Exercises 119122, determine whether each statement is true or fa...
 6.4.6.1.509: In Exercises 119122, determine whether each statement is true or fa...
 6.4.6.1.510: In Exercises 119122, determine whether each statement is true or fa...
 6.4.6.1.511: Where is the error in this proof that 2 0? a b Suppose that a and b...
 6.4.6.1.512: In Exercises 124127, factor each polynomial. x2 y2 3x 3y
 6.4.6.1.513: In Exercises 124127, factor each polynomial. x2n 25y2n
 6.4.6.1.514: In Exercises 124127, factor each polynomial. 4x2n 12xn 9
 6.4.6.1.515: In Exercises 124127, factor each polynomial. (x 3)2 2(x 3) 1
 6.4.6.1.516: In Exercises 128129, find all integers k so that the trinomial is a...
 6.4.6.1.517: In Exercises 128129, find all integers k so that the trinomial is a...
 6.4.6.1.518: In Exercises 130133, use the GRAPH or TABLE feature of a graphing u...
 6.4.6.1.519: In Exercises 130133, use the GRAPH or TABLE feature of a graphing u...
 6.4.6.1.520: In Exercises 130133, use the GRAPH or TABLE feature of a graphing u...
 6.4.6.1.521: In Exercises 130133, use the GRAPH or TABLE feature of a graphing u...
 6.4.6.1.522: Simplify: (2x2y3)4(5xy2). (Section 5.7, Example 5)
 6.4.6.1.523: Subtract: (10x2 5x 2) (14x2 5x 1). (Section 5.1, Example 3)
 6.4.6.1.524: Divide: 6x2 11x 10 3x 2 . (Section 5.6, Example 1)
 6.4.6.1.525: Exercises 137139 will help you prepare for the material covered in ...
 6.4.6.1.526: Exercises 137139 will help you prepare for the material covered in ...
 6.4.6.1.527: Exercises 137139 will help you prepare for the material covered in ...
Solutions for Chapter 6.4: Factoring Special Forms
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 6.4: Factoring Special Forms
Get Full SolutionsThis textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Chapter 6.4: Factoring Special Forms includes 154 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 154 problems in chapter 6.4: Factoring Special Forms have been answered, more than 71372 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.

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

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

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.

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).

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!).

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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 .

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.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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