- 6-2.1: 2a + 5b
- 6-2.2: 1 3 x3 - 9y
- 6-2.3: mw 2 - 3 nz3 + 1
- 6-2.4: (2a + 3b) + (8a - 5b)
- 6-2.5: (x2 - 4x + 3) - (4x2 + 3x - 5)
- 6-2.6: 2x(3y + 9)
- 6-2.7: 2p2q(5pq - 3p3q2 + 4pq4)
- 6-2.8: (y - 10)(y + 7)
- 6-2.9: (x + 6)(x + 3)
- 6-2.10: (2z - 1)(2z + 1)
- 6-2.11: (2m - 3n)2
- 6-2.12: (x + 1)(x2 - 2x + 3)
- 6-2.13: (2x - 1)(x2 - 4x + 4)
- 6-2.14: GEOMETRY Find the area of the triangle.
- 6-2.15: 3z2 - 5z + 11
- 6-2.16: x3 - 9
- 6-2.17: 6xy z - _3c d
- 6-2.18: m - 5 19.
- 6-2.19: 5x2y4 + x 3 2
- 6-2.20: 4 3 y2 + _5 6 y7
- 6-2.21: (3x2 - x + 2) + (x2 + 4x - 9)
- 6-2.22: (5y + 3y2) + (-8y - 6y2)
- 6-2.23: (9r2 + 6r + 16) - (8r2 + 7r + 10)
- 6-2.24: (7m2 + 5m - 9) + (3m2 - 6)
- 6-2.25: 4b(cb - zd)
- 6-2.26: 4a(3a2 + b)
- 6-2.27: -5ab2(-3a2b + 6a3b - 3a4b4)
- 6-2.28: 2xy(3xy3 - 4xy + 2y4)
- 6-2.29: (p + 6)(p - 4)
- 6-2.30: (a + 6)(a + 3)
- 6-2.31: (b + 5)(b - 5)
- 6-2.32: (6 - z)(6 + z)
- 6-2.33: (3x + 8)(2x + 6)
- 6-2.34: (4y - 6)(2y + 7)
- 6-2.35: (3b - c)3
- 6-2.36: (x2 + xy + y2)(x - y)
- 6-2.37: PERSONAL FINANCE Toshiro has $850 to invest. He can invest in a sav...
- 6-2.38: Write a polynomial to represent the profit generated by the product.
- 6-2.39: Find the profit from sales of 1850 units.
- 6-2.40: Simplify (c2 - 6cd - 2d2) + (7c2 - cd + 8d2) - (-c2 + 5cd - d2).
- 6-2.41: Find the product of x2 + 6x - 5 and -3x + 2.
- 6-2.42: (4x2 - 3y2 + 5xy) - (8xy + 3y2)
- 6-2.43: (10x2 - 3xy + 4y2) - (3x2 + 5xy)
- 6-2.44: 3 4 x2(8x + 12y - 16xy2)
- 6-2.45: _1 2 a3(4a - 6b + 8ab4)
- 6-2.46: d-3(d5 - 2d3 + d-1)
- 6-2.47: x-3y2(yx4 + y-1x3 + y-2x2)
- 6-2.48: (a3 - b)(a3 + b)
- 6-2.49: (m2 - 5)(2m2 + 3)
- 6-2.50: (x - 3y)2
- 6-2.51: (1 + 4c)2
- 6-2.52: GENETICS Suppose R and W represent two genes that a plant can inher...
- 6-2.53: OPEN ENDED Write a polynomial of degree 5 that has three terms.
- 6-2.54: Which One Doesnt Belong? Identify the expression that does not belo...
- 6-2.55: CHALLENGE What is the degree of the product of a polynomial of degr...
- 6-2.56: Writing in Math Use the information about tuition increases to expl...
- 6-2.57: ACT/SAT Which polynomial has degree 3? A x3 + x2 - 2x4 B -2x2 - 3x ...
- 6-2.58: REVIEW (-4x2 + 2x + 3) - 3(2x2 - 5x + 1) = F 2x2 G -10x2 H -10x2 + ...
- 6-2.59: (-4d2)3
- 6-2.60: 5rt2(2rt)2
- 6-2.61: x2yz _ 4 xy3z2
- 6-2.62: ( 3ab _2 6a2b) 2
- 6-2.63: y > x2 - 4x + 6
- 6-2.64: y -x2 + 6x - 3
- 6-2.65: y < x2 - 2x
- 6-2.66: f(x) = x2 - 8x + 3
- 6-2.67: f(x) = -3x2 - 18x + 5
- 6-2.68: f(x) = -7 + 4x2
- 6-2.69: A + D
- 6-2.70: B - C
- 6-2.71: 3B - 2A
- 6-2.72: y O x (1, 1) (3, 1)
- 6-2.73: y O x (2, 0) (4, 4) 7
- 6-2.74: In 1990, 2,573,225 people attended St. Louis Cardinals home games. ...
- 6-2.75: x_ 3 x
- 6-2.76: 4y5 _ 2y2
- 6-2.77: x2y3 _ xy
- 6-2.78: 9a3 _b 3ab
Solutions for Chapter 6-2: Operations with Polynomials
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
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!
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.)
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 n-l - An).
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
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 = M-I 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)j-I 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.