- 8.5.1: True or False The equation is an example of an identity.
- 8.5.2: True or False The rational expression is proper.
- 8.5.3: Factor completely: 3x4 + 6x3 + 3x2
- 8.5.4: True or False Every polynomial with real numbers as coefficients ca...
- 8.5.5: x x2 - 1
- 8.5.6: 5x + 2 x3 - 1
- 8.5.7: x2 + 5 x2 - 4
- 8.5.8: 3x2 - 2 x2 - 1
- 8.5.9: 5x3 + 2x - 1 x2 - 4
- 8.5.10: 3x4 + x2 - 2 x3 + 8
- 8.5.11: x1x - 12 1x + 421x - 32
- 8.5.12: 2x1x2 + 42 x2 + 1
- 8.5.13: 4 x1x - 12
- 8.5.14: 3x 1x + 221x - 12
- 8.5.15: 1 x1x2 + 12
- 8.5.16: 1 1x + 121x2 + 42
- 8.5.17: x 1x - 121x - 22
- 8.5.18: 3x 1x + 221x - 42
- 8.5.19: x2 1x - 122 1x + 1
- 8.5.20: x + 1 x2 1x - 22
- 8.5.21: 1 x3 - 8
- 8.5.22: 2x + 4 x3 - 1
- 8.5.23: x2 1x - 122 1x + 122
- 8.5.24: x + 1 x2 1x - 222
- 8.5.25: x - 3 1x + 221x + 122
- 8.5.26: x2 + x 1x + 221x - 122
- 8.5.27: x + 4 x2 1x2 + 42
- 8.5.28: 10x2 + 2x 1x - 122 1x2 + 22
- 8.5.29: x2 + 2x + 3 1x + 121x2 + 2x + 42
- 8.5.30: x2 - 11x - 18 x1x2 + 3x + 32
- 8.5.31: x 13x - 2212x + 12
- 8.5.32: 1 12x + 3214x - 12
- 8.5.33: x x2 + 2x - 3
- 8.5.34: x2 - x - 8 1x + 121x2 + 5x + 62
- 8.5.35: x2 + 2x + 3 1x2 + 42 2
- 8.5.36: x3 + 1 1x2 + 162 2
- 8.5.37: 7x + 3 x3 - 2x2 - 3x
- 8.5.38: x3 + 1 x5 - x4
- 8.5.39: x2 x3 - 4x2 + 5x - 2
- 8.5.40: x2 + 1 x3 + x2 - 5x + 3
- 8.5.41: x3 1x2 + 162 3
- 8.5.42: x2 1x2 + 42 3
- 8.5.43: 4 2x2 - 5x - 3
- 8.5.44: 4x 2x2 + 3x - 2
- 8.5.45: 2x + 3 x4 - 9x2
- 8.5.46: x2 + 9 x4 - 2x2 - 8
Solutions for Chapter 8.5: Partial Fraction Decomposition
Full solutions for College Algebra | 9th 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!
Column space C (A) =
space of all combinations of the columns of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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].
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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).
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.