 58.1: Sharon said that if f(x) is a polynomial function and f(a) 5 0, the...
 58.2: Jordan said that if the roots of a polynomial function f(x) are r1 ...
 58.3: In 318, find all roots of each given function by factoring or by us...
 58.4: In 318, find all roots of each given function by factoring or by us...
 58.5: In 318, find all roots of each given function by factoring or by us...
 58.6: In 318, find all roots of each given function by factoring or by us...
 58.7: In 318, find all roots of each given function by factoring or by us...
 58.8: In 318, find all roots of each given function by factoring or by us...
 58.9: In 318, find all roots of each given function by factoring or by us...
 58.10: In 318, find all roots of each given function by factoring or by us...
 58.11: In 318, find all roots of each given function by factoring or by us...
 58.12: In 318, find all roots of each given function by factoring or by us...
 58.13: In 318, find all roots of each given function by factoring or by us...
 58.14: In 318, find all roots of each given function by factoring or by us...
 58.15: In 318, find all roots of each given function by factoring or by us...
 58.16: In 318, find all roots of each given function by factoring or by us...
 58.17: In 318, find all roots of each given function by factoring or by us...
 58.18: In 318, find all roots of each given function by factoring or by us...
 58.19: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.20: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.21: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.22: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.23: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.24: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.25: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.26: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.27: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.28: In 1928: a. Find f(a) for each given function. b. Is a a root of th...
 58.29: a. Verify by multiplication that (x 2 1)(x2 1 x 1 1) 5 x3 2 1. b. U...
 58.30: a. Verify by multiplication that (x 1 1)(x2 2 x 1 1) 5 x3 1 1. b. U...
 58.31: Let f(x) 5 x3 1 3x2 2 2x 2 6 and g(x) 5 2f(x) 5 2x3 1 6x 2 4x 212 b...
Solutions for Chapter 58: Solving Higher Degree Polynomial Equations
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 58: Solving Higher Degree Polynomial Equations
Get Full SolutionsSince 31 problems in chapter 58: Solving Higher Degree Polynomial Equations have been answered, more than 30775 students have viewed full stepbystep solutions from this chapter. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. Chapter 58: Solving Higher Degree Polynomial Equations includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.