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# Solutions for Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems

## Full solutions for College Algebra | 6th Edition

ISBN: 9780321782281

Solutions for Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems

Solutions for Chapter 3.3
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##### ISBN: 9780321782281

This textbook survival guide was created for the textbook: College Algebra , edition: 6. Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems includes 81 full step-by-step solutions. College Algebra was written by and is associated to the ISBN: 9780321782281. Since 81 problems in chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems have been answered, more than 37082 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Cofactor Cij.

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

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• 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.

• Kronecker product (tensor product) A ® B.

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

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• 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 .

• 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).

• Multiplication Ax

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

• Pivot columns of A.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• 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.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B II·

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

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