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Solutions for Chapter 6.4: Division of Polynomials

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

ISBN: 9780321758934

Solutions for Chapter 6.4: Division of Polynomials

Solutions for Chapter 6.4
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Textbook: Intermediate Algebra for College Students
Edition: 6
Author: Robert F. Blitzer
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. Chapter 6.4: Division of Polynomials includes 86 full step-by-step solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 86 problems in chapter 6.4: Division of Polynomials have been answered, more than 53005 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Companion matrix.

    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).

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • Determinant IAI = det(A).

    Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • 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.

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • 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.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Schwarz inequality

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

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Singular matrix A.

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

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • 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.