 5.5.1: Describe a use for the Remainder Theorem.
 5.5.2: Explain why the Rational Zero Theorem does not guarantee finding ze...
 5.5.3: What is the difference between rational and real zeros?
 5.5.4: . If Descartes Rule of Signs reveals a no change of signs or one si...
 5.5.5: If synthetic division reveals a zero, why should we try that value ...
 5.5.6: For the following exercises, use the Remainder Theorem to find the ...
 5.5.7: For the following exercises, use the Remainder Theorem to find the ...
 5.5.8: For the following exercises, use the Remainder Theorem to find the ...
 5.5.9: For the following exercises, use the Remainder Theorem to find the ...
 5.5.10: For the following exercises, use the Remainder Theorem to find the ...
 5.5.11: For the following exercises, use the Remainder Theorem to find the ...
 5.5.12: For the following exercises, use the Remainder Theorem to find the ...
 5.5.13: For the following exercises, use the Remainder Theorem to find the ...
 5.5.14: For the following exercises, use the Factor Theorem to find all rea...
 5.5.15: For the following exercises, use the Factor Theorem to find all rea...
 5.5.16: For the following exercises, use the Factor Theorem to find all rea...
 5.5.17: For the following exercises, use the Factor Theorem to find all rea...
 5.5.18: For the following exercises, use the Factor Theorem to find all rea...
 5.5.19: For the following exercises, use the Factor Theorem to find all rea...
 5.5.20: For the following exercises, use the Factor Theorem to find all rea...
 5.5.21: For the following exercises, use the Factor Theorem to find all rea...
 5.5.22: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.23: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.24: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.25: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.26: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.27: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.28: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.29: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.30: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.31: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.32: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.33: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.34: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.35: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.36: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.37: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.38: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.39: For the following exercises, use the Rational Zero Theorem to find ...
 5.5.40: For the following exercises, find all complex solutions (real and n...
 5.5.41: For the following exercises, find all complex solutions (real and n...
 5.5.42: For the following exercises, find all complex solutions (real and n...
 5.5.43: For the following exercises, find all complex solutions (real and n...
 5.5.44: For the following exercises, find all complex solutions (real and n...
 5.5.45: For the following exercises, find all complex solutions (real and n...
 5.5.46: Use Descartes Rule to determine the possible number of positive and...
 5.5.47: Use Descartes Rule to determine the possible number of positive and...
 5.5.48: Use Descartes Rule to determine the possible number of positive and...
 5.5.49: Use Descartes Rule to determine the possible number of positive and...
 5.5.50: Use Descartes Rule to determine the possible number of positive and...
 5.5.51: Use Descartes Rule to determine the possible number of positive and...
 5.5.52: Use Descartes Rule to determine the possible number of positive and...
 5.5.53: Use Descartes Rule to determine the possible number of positive and...
 5.5.54: Use Descartes Rule to determine the possible number of positive and...
 5.5.55: Use Descartes Rule to determine the possible number of positive and...
 5.5.56: For the following exercises, list all possible rational zeros for t...
 5.5.57: For the following exercises, list all possible rational zeros for t...
 5.5.58: For the following exercises, list all possible rational zeros for t...
 5.5.59: For the following exercises, list all possible rational zeros for t...
 5.5.60: For the following exercises, list all possible rational zeros for t...
 5.5.61: For the following exercises, use your calculator to graph the polyn...
 5.5.62: For the following exercises, use your calculator to graph the polyn...
 5.5.63: For the following exercises, use your calculator to graph the polyn...
 5.5.64: For the following exercises, use your calculator to graph the polyn...
 5.5.65: For the following exercises, use your calculator to graph the polyn...
 5.5.66: For the following exercises, construct a polynomial function of lea...
 5.5.67: For the following exercises, construct a polynomial function of lea...
 5.5.68: For the following exercises, construct a polynomial function of lea...
 5.5.69: For the following exercises, construct a polynomial function of lea...
 5.5.70: For the following exercises, find the dimensions of the box describ...
 5.5.71: For the following exercises, find the dimensions of the box describ...
 5.5.72: For the following exercises, find the dimensions of the box describ...
 5.5.73: For the following exercises, find the dimensions of the box describ...
 5.5.74: For the following exercises, find the dimensions of the box describ...
 5.5.75: For the following exercises, find the dimensions of the box describ...
 5.5.76: For the following exercises, find the dimensions of the right circu...
 5.5.77: For the following exercises, find the dimensions of the right circu...
 5.5.78: For the following exercises, find the dimensions of the right circu...
 5.5.79: For the following exercises, find the dimensions of the right circu...
 5.5.80: For the following exercises, find the dimensions of the right circu...
Solutions for Chapter 5.5: ZEROS OF POLYNOMIAL FUNCTIONS
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 5.5: ZEROS OF POLYNOMIAL FUNCTIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 1. Since 80 problems in chapter 5.5: ZEROS OF POLYNOMIAL FUNCTIONS have been answered, more than 34503 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781938168383. Chapter 5.5: ZEROS OF POLYNOMIAL FUNCTIONS includes 80 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Outer product uv T
= column times row = rank one matrix.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Stiffness matrix
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.

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.