×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide

Already have an account? Login here
×
Reset your password

Solutions for Chapter 8.5: Polynomial and Rational Inequalities

Full solutions for Intermediate Algebra for College Students | 6th Edition

ISBN: 9780321758934

Solutions for Chapter 8.5: Polynomial and Rational Inequalities

Solutions for Chapter 8.5
4 5 0 368 Reviews
14
5
Textbook: Intermediate Algebra for College Students
Edition: 6
Author: Robert F. Blitzer
ISBN: 9780321758934

Chapter 8.5: Polynomial and Rational Inequalities includes 111 full step-by-step solutions. Since 111 problems in chapter 8.5: Polynomial and Rational Inequalities have been answered, more than 53326 students have viewed full step-by-step solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • 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.

  • 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.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • 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).

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Rotation matrix

    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 ().

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Singular Value Decomposition

    (SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

  • 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.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • 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.