 77.1: Find each product. (a + 6) 2
 77.2: Find each product. (2a + 7b) 2
 77.3: Find each product. (3x + 9y) 2
 77.4: Find each product. (4n  3)(4n 3)
 77.5: Find each product. (x 2  6y) 2
 77.6: Find each product. (9  p) 2
 77.7: GENETICS For Exercises 7 and 8, use the following information. Dali...
 77.8: GENETICS For Exercises 7 and 8, use the following information. Dali...
 77.9: Find each product. (8x  5)(8x + 5)
 77.10: Find each product. (3a + 7b)(3a  7b)
 77.11: Find each product. (4y 2 + 3z)(4y 2  3z)
 77.12: Find each product. (k + 8)(k + 8)
 77.13: Find each product. (y + 4) 2
 77.14: Find each product. (2g + 5) 2
 77.15: Find each product. (a  5)(a 5)
 77.16: Find each product. (n  12) 2
 77.17: Find each product. (7  4y) 2
 77.18: GENETICS For Exercises 18 and 19, use the following information and...
 77.19: GENETICS For Exercises 18 and 19, use the following information and...
 77.20: Find each product. (b + 7)(b  7)
 77.21: Find each product. (c  2)(c + 2)
 77.22: Find each product. (11r + 8)(11r  8)
 77.23: Find each product. (9x + 3) 2
 77.24: Find each product. (4  6h) 2
 77.25: Find each product. (12p  3)(12p + 3)
 77.26: Find each product. (a + 5b) 2
 77.27: Find each product. (m + 7n) 2
 77.28: Find each product. (2x  9y) 2
 77.29: Find each product. (3n  10p) 2
 77.30: Find each product. (5w + 14)(5w 14)
 77.31: Find each product. (4d  13)(4d + 13)
 77.32: Find each product. ( x 3 + 4y) 2
 77.33: Find each product. (3 a 2  b 2 ) 2
 77.34: Find each product. (8 a2  9 b3 )(8 a 2 + 9 b3 )
 77.35: Find each product. (5 x4  y)(5 x4 + y)
 77.36: Find each product. (_2 3 x  6) 2
 77.37: Find each product. (_45 x + 10) 2
 77.38: Find each product. (2n + 1)(2n  1)(n + 5)
 77.39: Find each product. (p + 3)(p 4)(p  3)(p + 4)
 77.40: MAGIC TRICK For Exercises 4043, use the following information. Madi...
 77.41: MAGIC TRICK For Exercises 4043, use the following information. Madi...
 77.42: MAGIC TRICK For Exercises 4043, use the following information. Madi...
 77.43: MAGIC TRICK For Exercises 4043, use the following information. Madi...
 77.44: WRESTLING For Exercises 4446, use the following information. A high...
 77.45: WRESTLING For Exercises 4446, use the following information. A high...
 77.46: WRESTLING For Exercises 4446, use the following information. A high...
 77.47: GEOMETRY The area of the shaded region models the difference of two...
 77.48: REASONING Compare and contrast the pattern for the square of a sum ...
 77.49: ALGEBRA TILES Draw a diagram to show how you would use algebra tile...
 77.50: OPEN ENDED Write two binomials whose product is a difference of squ...
 77.51: CHALLENGE Does a pattern exist for the cube of a sum, (a + b ) 3 ? ...
 77.52: Writing in Math Using the information about the product of two bino...
 77.53: The base of a triangle is represented by x  4, and the height is r...
 77.54: REVIEW The sum of a number and 8 is 19. Which equation shows this ...
 77.55: Find each product. (x + 2)(x + 7)
 77.56: Find each product. (c  9)(c + 3)
 77.57: Find each product. (4y  1)(5y  6)
 77.58: Find each product. (3n  5)(8n + 5)
 77.59: Find each product. (x  2)(3 x 2  5x + 4)
 77.60: Find each product. (2k + 5)(2 k 2  8k + 7)
 77.61: Solve. 6(x + 2) + 4 = 5(3x  4)
 77.62: Solve. 3(3a  8) + 2a = 4(2a + 1)
 77.63: Solve. p(p + 2) + 3p = p(p  3)
 77.64: Solve. y(y  4) + 2y = y(y + 12)  7
 77.65: Use elimination to solve each system of equations. _3 4 x + _1 5 y = 5
 77.66: Use elimination to solve each system of equations. 2x  y = 10
 77.67: Use elimination to solve each system of equations. 2x = 4  3y
 77.68: Write the slopeintercept form of an equation that passes through t...
 77.69: Write the slopeintercept form of an equation that passes through t...
 77.70: Write the slopeintercept form of an equation that passes through t...
 77.71: Find the nth term of each arithmetic sequence described. a1 = 3, d ...
 77.72: Find the nth term of each arithmetic sequence described. 5, 1, 7, ...
 77.73: PHYSICAL FITNESS Mitchell likes to exercise regularly. He likes to ...
Solutions for Chapter 77: Special Products
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 77: Special Products
Get Full SolutionsChapter 77: Special Products includes 73 full stepbystep solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This expansive textbook survival guide covers the following chapters and their solutions. Since 73 problems in chapter 77: Special Products have been answered, more than 35536 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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).

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.

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).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.