 3.3.1: Two forms of the Division Algorithm are shown below. Identify and l...
 3.3.2: The rational expression is called ________ if the degree of the num...
 3.3.3: In the Division Algorithm, the rational expression is ________ beca...
 3.3.4: An alternative method to long division of polynomials is called ___...
 3.3.5: The ________ Theorem states that a polynomial has a factor if and o...
 3.3.6: The ________ Theorem states that if a polynomial is divided by the ...
 3.3.7:
 3.3.8: y2 x 2 8 39 x2 5 y
 3.3.9: y1 x2 2x 1 x 3 ,
 3.3.10: y1 x4 x2 1 x2 1 ,
 3.3.11: 2x2 10x 12 x 3
 3.3.12: 5x2 17x 12 x 4 2
 3.3.13: 4x3 7x 2 11x 5 4x 5
 3.3.14: 6x3 16x 2 17x 6 3x 2 4
 3.3.15: x4 5x3 6x2 x 2 x 2
 3.3.16: x3 4x 2 3x 12 x 3 x4
 3.3.17: x 3 125 x 5 3 27 x 3 x
 3.3.18: x x 3 125 x 5
 3.3.19: 7x 3 x 2
 3.3.20: 8x 5 2x 1
 3.3.21: x 3 1 3 9 x2 1
 3.3.22: x5 7 x x 3 1 3
 3.3.23: 3x 2x3 9 8x2 x2 1
 3.3.24: 5x3 16 20x x4 x2 x 3 3x
 3.3.25: x 4 x 13
 3.3.26: 2x3 4x2 15x 5 x 12 x
 3.3.27: 3x3 17x2 15x 25 x 5 2x
 3.3.28: 5x3 18x2 7x 6 x 3
 3.3.29: 6x3 7x2 x 26 x 3
 3.3.30: 2x3 14x2 20x 7 x 6
 3.3.31: 4x3 9x 8x2 18 x 2
 3.3.32: 9x3 16x 18x2 32 x 2 4
 3.3.33: x3 75x 250 x 10
 3.3.34: 3x3 16x2 72 x 6 x
 3.3.35: 5x3 6x2 8 x 4
 3.3.36: 5x3 6x 8 x 2
 3.3.37: 10x4 50x3 800 x 6 5
 3.3.38: x5 13x4 120x 80 x 3 1
 3.3.39: x3 512 x 8
 3.3.40: x 3 729 x 9 x
 3.3.41: 3x4 x 2 x
 3.3.42: 3x 4 x 2
 3.3.43: 180x x4 x 6 3
 3.3.44: 5 3x 2x2 x3 x 1 18
 3.3.45: 4x3 16x2 23x 15 x 1 2 5
 3.3.46: 3x3 4x2 5 x 3 2 4
 3.3.47: f x x k 4
 3.3.48: f x x k 2 3
 3.3.49: f x 15x 4 10x3 6x2 14,
 3.3.50: f x 10x3 22x2 3x 4,
 3.3.51: f x x k 2 3 3x2 2x 14, k
 3.3.52: f x x k 5 3 2x2 5x 4, f
 3.3.53: f x 4x k 1 3 3 6x2 12x 4, f
 3.3.54: f x 3x k 2 2 3 8x2 10x 8, f
 3.3.55: f f 2 1 2 f 1 f 2 f x
 3.3.56: 1 g 3 g 1 g x
 3.3.57: h 2 h 2 h 5 h x
 3.3.58: f 1 f 2 f 5 f 10 f x
 3.3.59: x x 2 3 7x 6 0,
 3.3.60: x x 4 3 28x 48 0, x
 3.3.61: 2 2 x3 15x 2 27x 10 0, x
 3.3.62: 48 3 x3 80x 2 41x 6 0,
 3.3.63: x x 3 3 2x 2 3x 6 0, x
 3.3.64: x x 2 3 2x 2 2x 4 0, x
 3.3.65: x x 1 3 3 3x 2 2 0, x
 3.3.66: x x 2 5 3 x 2 13x 3 0, x x
 3.3.67: f x 2x x 2, x 1 3 x2 5x 2 f
 3.3.68: f x 3x x 3, x 2 3 2x2 19x 6 f
 3.3.69: x x x 5, x 4 4 4x3 15x2 f
 3.3.70: f x 8x x 2, x 4 4
 3.3.71: f x 6x 2x 1, 3x 2 3 41x2 9x 14 10
 3.3.72: f x 10x 2x 5, 5x 3 3 11x2 72x 45 f
 3.3.73: f x 2x 2x 1, x 5 3 x2 10x 5 f x
 3.3.74: f x x x 43, x 3 3 3x2 48x 144 f
 3.3.75: f x x3 2x2 5x 10 f
 3.3.76: g x x3 4x2 2x 8 f
 3.3.77: h t t3 2t 2 7t 2 g
 3.3.78: f s s3 12s2 40s 24 h
 3.3.79: h x x5 7x4 10x3 14x2 24x f
 3.3.80: g x 6x4 11x3 51x2 99x 27 h
 3.3.81: 4x3 8x2 x 3 2x 3 g
 3.3.82: 3 x2 64x 64 x 8 4
 3.3.83: x4 6x3 11x 2 6x x2 3x 2
 3.3.84: x4 9x3 5x 2 36x 4 x2 4 x
 3.3.85: DATA ANALYSIS: HIGHER EDUCATION The amounts (in billions of dollars...
 3.3.86: DATA ANALYSIS: HEALTH CARE The amounts (in billions of dollars) of ...
 3.3.87: If is a factor of some polynomial function then is a zero o
 3.3.88: 6x 6 x 5 92x 4 45x 3 184x2 4x 48. 2x
 3.3.89: The rational expression is improper
 3.3.90: Use the form to create a cubic function that (a) passes through the...
 3.3.91: x3n 9x2n 27xn 27 xn 3
 3.3.92: x3n 3x2n 5xn 6 x n 2 x3
 3.3.93: WRITING Briefly explain what it means for a divisor to divide evenl...
 3.3.94: WRITING Briefly explain how to check polynomial division, and justi...
 3.3.95: x3 4x2 3x c x 5 c
 3.3.96: x5 2x2 x c x 2
 3.3.97: THINK ABOUT IT Find the value of such that is a factor of
 3.3.98: THINK ABOUT IT Find the value of such that is a factor of
 3.3.99: WRITING Complete each polynomial division. Write a brief descriptio...
 3.3.100: CAPSTONE Consider the division where (a) What is the remainder when...
Solutions for Chapter 3.3: Polynomial and Synthetic Division
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 3.3: Polynomial and Synthetic Division
Get Full SolutionsChapter 3.3: Polynomial and Synthetic Division includes 100 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. Since 100 problems in chapter 3.3: Polynomial and Synthetic Division have been answered, more than 47336 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.