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Solutions for Chapter 3.4: Zeros of Polynomial Functions

College Algebra | 7th Edition | ISBN: 9780134469164 | Authors: Robert F. Blitzer

Full solutions for College Algebra | 7th Edition

ISBN: 9780134469164

College Algebra | 7th Edition | ISBN: 9780134469164 | Authors: Robert F. Blitzer

Solutions for Chapter 3.4: Zeros of Polynomial Functions

Solutions for Chapter 3.4
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Textbook: College Algebra
Edition: 7
Author: Robert F. Blitzer
ISBN: 9780134469164

Since 103 problems in chapter 3.4: Zeros of Polynomial Functions have been answered, more than 30845 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 7. College Algebra was written by and is associated to the ISBN: 9780134469164. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.4: Zeros of Polynomial Functions includes 103 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

    The n roots are the eigenvalues of A.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Column space C (A) =

    space of all combinations of the columns of A.

  • 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.

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Eigenvalue A and eigenvector x.

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

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • 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.

  • 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.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Schwarz inequality

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Vector v in Rn.

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

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