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Solutions for Chapter R.5 : Factoring Polynomials

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter R.5 : Factoring Polynomials

Solutions for Chapter R.5
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Textbook: College Algebra
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716811

This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321716811. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Chapter R.5 : Factoring Polynomials includes 136 full step-by-step solutions. Since 136 problems in chapter R.5 : Factoring Polynomials have been answered, more than 7961 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • 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).

  • Complex conjugate

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

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • 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.

  • 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.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Elimination.

    A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Hypercube matrix pl.

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

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • 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.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

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