 62.1: 2a + 5b
 62.2: 1 3 x3  9y
 62.3: mw 2  3 nz3 + 1
 62.4: (2a + 3b) + (8a  5b)
 62.5: (x2  4x + 3)  (4x2 + 3x  5)
 62.6: 2x(3y + 9)
 62.7: 2p2q(5pq  3p3q2 + 4pq4)
 62.8: (y  10)(y + 7)
 62.9: (x + 6)(x + 3)
 62.10: (2z  1)(2z + 1)
 62.11: (2m  3n)2
 62.12: (x + 1)(x2  2x + 3)
 62.13: (2x  1)(x2  4x + 4)
 62.14: GEOMETRY Find the area of the triangle.
 62.15: 3z2  5z + 11
 62.16: x3  9
 62.17: 6xy z  _3c d
 62.18: m  5 19.
 62.19: 5x2y4 + x 3 2
 62.20: 4 3 y2 + _5 6 y7
 62.21: (3x2  x + 2) + (x2 + 4x  9)
 62.22: (5y + 3y2) + (8y  6y2)
 62.23: (9r2 + 6r + 16)  (8r2 + 7r + 10)
 62.24: (7m2 + 5m  9) + (3m2  6)
 62.25: 4b(cb  zd)
 62.26: 4a(3a2 + b)
 62.27: 5ab2(3a2b + 6a3b  3a4b4)
 62.28: 2xy(3xy3  4xy + 2y4)
 62.29: (p + 6)(p  4)
 62.30: (a + 6)(a + 3)
 62.31: (b + 5)(b  5)
 62.32: (6  z)(6 + z)
 62.33: (3x + 8)(2x + 6)
 62.34: (4y  6)(2y + 7)
 62.35: (3b  c)3
 62.36: (x2 + xy + y2)(x  y)
 62.37: PERSONAL FINANCE Toshiro has $850 to invest. He can invest in a sav...
 62.38: Write a polynomial to represent the profit generated by the product.
 62.39: Find the profit from sales of 1850 units.
 62.40: Simplify (c2  6cd  2d2) + (7c2  cd + 8d2)  (c2 + 5cd  d2).
 62.41: Find the product of x2 + 6x  5 and 3x + 2.
 62.42: (4x2  3y2 + 5xy)  (8xy + 3y2)
 62.43: (10x2  3xy + 4y2)  (3x2 + 5xy)
 62.44: 3 4 x2(8x + 12y  16xy2)
 62.45: _1 2 a3(4a  6b + 8ab4)
 62.46: d3(d5  2d3 + d1)
 62.47: x3y2(yx4 + y1x3 + y2x2)
 62.48: (a3  b)(a3 + b)
 62.49: (m2  5)(2m2 + 3)
 62.50: (x  3y)2
 62.51: (1 + 4c)2
 62.52: GENETICS Suppose R and W represent two genes that a plant can inher...
 62.53: OPEN ENDED Write a polynomial of degree 5 that has three terms.
 62.54: Which One Doesnt Belong? Identify the expression that does not belo...
 62.55: CHALLENGE What is the degree of the product of a polynomial of degr...
 62.56: Writing in Math Use the information about tuition increases to expl...
 62.57: ACT/SAT Which polynomial has degree 3? A x3 + x2  2x4 B 2x2  3x ...
 62.58: REVIEW (4x2 + 2x + 3)  3(2x2  5x + 1) = F 2x2 G 10x2 H 10x2 + ...
 62.59: (4d2)3
 62.60: 5rt2(2rt)2
 62.61: x2yz _ 4 xy3z2
 62.62: ( 3ab _2 6a2b) 2
 62.63: y > x2  4x + 6
 62.64: y x2 + 6x  3
 62.65: y < x2  2x
 62.66: f(x) = x2  8x + 3
 62.67: f(x) = 3x2  18x + 5
 62.68: f(x) = 7 + 4x2
 62.69: A + D
 62.70: B  C
 62.71: 3B  2A
 62.72: y O x (1, 1) (3, 1)
 62.73: y O x (2, 0) (4, 4) 7
 62.74: In 1990, 2,573,225 people attended St. Louis Cardinals home games. ...
 62.75: x_ 3 x
 62.76: 4y5 _ 2y2
 62.77: x2y3 _ xy
 62.78: 9a3 _b 3ab
Solutions for Chapter 62: Operations with Polynomials
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 62: Operations with Polynomials
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Since 78 problems in chapter 62: Operations with Polynomials have been answered, more than 61378 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 62: Operations with Polynomials includes 78 full stepbystep solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302.

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!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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

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.

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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}.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.