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 5.6.5.1.716: In Exercises 5968, divide as indicated. x4 y4 x y
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 5.6.5.1.718: In Exercises 5968, divide as indicated. 3x4 5x3 7x2 3x 2 x2 x 2
 5.6.5.1.719: In Exercises 5968, divide as indicated. x4 x3 7x2 7x 2 x2 3x 2
 5.6.5.1.720: In Exercises 5968, divide as indicated. 4x3 3x2 x 1 x2 x 1
 5.6.5.1.721: In Exercises 5968, divide as indicated. x4 x2 1 x2 x 1
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 5.6.5.1.723: In Exercises 5968, divide as indicated. 5x5 7x4 3x3 20x2 28x 12 x3 4
 5.6.5.1.724: In Exercises 5968, divide as indicated. 4x3 7x2y 16xy2 3y3 x 3y
 5.6.5.1.725: In Exercises 5968, divide as indicated. 12x3 19x2y 13xy2 10y3 4x 5y
 5.6.5.1.726: Divide the difference between 4x3 x2 2x 7 and 3x3 2x2 7x 4 by x 1.
 5.6.5.1.727: Divide the difference between 4x3 2x2 x 1 and 2x3 x2 2x 5 by x 2.
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 5.6.5.1.745: When a certain polynomial is divided by 2x 4, the quotient is x 3 1...
 5.6.5.1.746: Find the number k such that when 16x2 2x k is divided by 2x 1, the ...
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 5.6.5.1.748: In Exercises 9195, use a graphing utility to determine whether the ...
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 5.6.5.1.753: Solve the system: 7x 6y 17 3x y 18. (Section 4.3, Example 2)
 5.6.5.1.754: What is 6% of 20? (Section 2.4, Example 5)
 5.6.5.1.755: Solve: x 3 2 5 x 5 2 5 . (Section 2.3, Example 4)
 5.6.5.1.756: Exercises 99101 will help you prepare for the material covered in t...
 5.6.5.1.757: Exercises 99101 will help you prepare for the material covered in t...
 5.6.5.1.758: Exercises 99101 will help you prepare for the material covered in t...
Solutions for Chapter 5.6: Long Division of Polynomials; Synthetic Division
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 5.6: Long Division of Polynomials; Synthetic Division
Get Full SolutionsIntroductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Chapter 5.6: Long Division of Polynomials; Synthetic Division includes 109 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Since 109 problems in chapter 5.6: Long Division of Polynomials; Synthetic Division have been answered, more than 67216 students have viewed full stepbystep solutions from this chapter.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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).

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

GramSchmidt 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.

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).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.