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# Solutions for Chapter 2.2: Polynomial Functions of Higher Degree

## Full solutions for Precalculus With Limits A Graphing Approach | 5th Edition

ISBN: 9780618851522

Solutions for Chapter 2.2: Polynomial Functions of Higher Degree

Solutions for Chapter 2.2
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##### ISBN: 9780618851522

This expansive textbook survival guide covers the following chapters and their solutions. Since 117 problems in chapter 2.2: Polynomial Functions of Higher Degree have been answered, more than 101915 students have viewed full step-by-step solutions from this chapter. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 2.2: Polynomial Functions of Higher Degree includes 117 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• 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.

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• 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.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• 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.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• 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.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Linearly dependent VI, ... , Vn.

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

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Right inverse A+.

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

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.