 66.1: 12x2  6x
 66.2: a2 + 5a + ab
 66.3: 21  7y + 3x  xy
 66.4: y2 + 4y + 2y + 8
 66.5: z2  4z  12
 66.6: 3b2  48
 66.7: 16w2  169
 66.8: h3 + 8000
 66.9: 5y4 + 7y3  8
 66.10: 84n4  62n2
 66.11: x4  50x2 + 49 = 0
 66.12: x3  125 = 0
 66.13: POOL The Shelby University swimming pool is in the shape of a recta...
 66.14: 2xy3  10x
 66.15: 6a2b2 + 18ab3
 66.16: 12cd3  8c2d2 + 10c5d3
 66.17: 3a2bx + 15cx2y + 25ad3y
 66.18: 8yz  6z  12y + 9
 66.19: 3ax  15a + x  5
 66.20: y2  5y + 4
 66.21: 2b2 + 13b  7
 66.22: z3 + 125
 66.23: t3  8
 66.24: 2x4 + 6x2  10
 66.25: a8 + 10a2  16
 66.26: 11n6 + 44n3
 66.27: 7b5  4b3 + 2b
 66.28: 7 x _1 9  3 x _1 3 + 4
 66.29: 6 x _2 5  4 x _1 5  16
 66.30: x4  34x2 + 225 = 0
 66.31: x4  15x2  16 = 0
 66.32: x4 + 6x2  27 = 0
 66.33: x3 + 64 = 0
 66.34: 27x3 + 1 = 0
 66.35: 8x3  27 = 0
 66.36: Write a polynomial equation that models the area of the frame.
 66.37: What are the dimensions of the glass piece?
 66.38: What are the dimensions of the frame?
 66.39: GEOMETRY The width of a rectangular prism is w centimeters. The hei...
 66.40: Find the factorization of 3x2 + x  2
 66.41: What are the factors of 2y2 + 9y + 4?
 66.42: 3n2 + 21n  24
 66.43: y4  z2
 66.44: 16a2 + 25b2
 66.45: 3x2  27y2
 66.46: x4  81
 66.47: 3a3 + 2a2  5a + 9a2b + 6ab  15b
 66.48: Factor the equation for the volume of the new piece to determine th...
 66.49: How much did each dimension of the packaging increase for the new f...
 66.50: LANDSCAPING A boardwalk that is x feet wide is built around a recta...
 66.51: 3x2 + 5x + 2 (3x + 2)(x + 1)
 66.52: x3 + 8 (x + 2)(x2  x + 4)
 66.53: 2x2  5x  3 (x  1)(2x + 3)
 66.54: 3x2  48 3(x + 4)(x  4)
 66.55: OPEN ENDED Give an example of an equation that is not quadratic but...
 66.56: CHALLENGE Factor 64p2n + 16pn + 1.
 66.57: REASONING Find a counterexample to the statement a2 + b2 = (a + b)2.
 66.58: CHALLENGE Explain how you would solve a  3 2  9a  3 = 8. Th...
 66.59: Writing in Math Use the information on page 349 to explain how solv...
 66.60: ACT/SAT Which is not a factor of x3  x2  2x? A x B x + 1 C x  1 ...
 66.61: ACT/SAT The measure of the largest angle of a triangle is 14 less t...
 66.62: REVIEW 27x3 + y3 = A (3x + y)(3x + y)(3x + y) B (3x + y)(9x2  3xy ...
 66.63: f(x) = x3  6x2 + 4x + 3
 66.64: f(x) = x4 + 2x3 + 3x2  7x + 4
 66.65: p(x) = x2  5x + 3
 66.66: p(x) = x3  11x  4
 66.67: p(x) = _2 3 x4  3x3
 66.68: PHOTOGRAPHY The perimeter of a rectangular picture is 86 inches. Tw...
 66.69: Determine whether each relation is a function. Write yes or no
 66.70: Determine whether each relation is a function. Write yes or no
 66.71: (x3 + 4x2  9x + 4) (x  1)
 66.72: (4x3  8x2  5x  10) (x + 2)
 66.73: (x4  9x2  2x + 6) (x  3)
 66.74: (x4 + 3x3  8x2 + 5x  6) (x + 1)
Solutions for Chapter 66: Solving Polynomial Equations
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 66: Solving Polynomial Equations
Get Full SolutionsChapter 66: Solving Polynomial Equations includes 74 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Since 74 problems in chapter 66: Solving Polynomial Equations have been answered, more than 55982 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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).

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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!

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.

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