 3.3.1: Fill in each blank so that the resulting statement is true.Consider...
 3.3.2: Fill in each blank so that the resulting statement is true.Consider...
 3.3.3: Fill in each blank so that the resulting statement is true.In the f...
 3.3.4: Fill in each blank so that the resulting statement is true.In the f...
 3.3.5: Fill in each blank so that the resulting statement is true.In the f...
 3.3.6: Fill in each blank so that the resulting statement is true.After pe...
 3.3.7: Fill in each blank so that the resulting statement is true.To divid...
 3.3.8: Fill in each blank so that the resulting statement is true.To divid...
 3.3.9: Fill in each blank so that the resulting statement is true.True or ...
 3.3.10: Fill in each blank so that the resulting statement is true.The Rema...
 3.3.11: Fill in each blank so that the resulting statement is true.The Fact...
 3.3.12: In Exercises 116, divide using long division. State the quotient,q(...
 3.3.13: In Exercises 116, divide using long division. State the quotient,q(...
 3.3.14: In Exercises 116, divide using long division. State the quotient,q(...
 3.3.15: In Exercises 116, divide using long division. State the quotient,q(...
 3.3.16: In Exercises 116, divide using long division. State the quotient,q(...
 3.3.17: In Exercises 1732, divide using synthetic division.(2x2 + x  10) ,...
 3.3.18: In Exercises 1732, divide using synthetic division.(x2 + x  2) , (...
 3.3.19: In Exercises 1732, divide using synthetic division.(3x2 + 7x  20) ...
 3.3.20: In Exercises 1732, divide using synthetic division.(5x2  12x  8) ...
 3.3.21: In Exercises 1732, divide using synthetic division.(4x3  3x2 + 3x ...
 3.3.22: In Exercises 1732, divide using synthetic division.(5x3  6x2 + 3x ...
 3.3.23: In Exercises 1732, divide using synthetic division.(6x5  2x3 + 4x2...
 3.3.24: In Exercises 1732, divide using synthetic division.(x5 + 4x4  3x2 ...
 3.3.25: In Exercises 1732, divide using synthetic division.(x2  5x  5x3 +...
 3.3.26: In Exercises 1732, divide using synthetic division.x2  6x  6x3 + ...
 3.3.27: In Exercises 1732, divide using synthetic division.x5 + x3  2x  1
 3.3.28: In Exercises 1732, divide using synthetic division.x7 + x5  10x3 +...
 3.3.29: In Exercises 1732, divide using synthetic division.x4  256x  4
 3.3.30: In Exercises 1732, divide using synthetic division.x7  128x  2
 3.3.31: In Exercises 1732, divide using synthetic division.2x5  3x4 + x3 ...
 3.3.32: In Exercises 1732, divide using synthetic division.x5  2x4  x3 + ...
 3.3.33: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.34: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.35: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.36: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.37: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.38: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.39: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.40: In Exercises 3340, use synthetic division and the RemainderTheorem ...
 3.3.41: Use synthetic division to dividef(x) = x3  4x2 + x + 6 by x + 1.Us...
 3.3.42: Use synthetic division to dividef(x) = x3  2x2  x + 2 by x + 1.Us...
 3.3.43: Solve the equation 2x3  5x2 + x + 2 = 0 given that 2 is azero of f...
 3.3.44: Solve the equation 2x3  3x2  11x + 6 = 0 given that 2is a zero o...
 3.3.45: Solve the equation 12x3 + 16x2  5x  3 = 0 given that  32is a root.
 3.3.46: Solve the equation 3x3 + 7x2  22x  8 = 0 given that  13is a root.
 3.3.47: In Exercises 4750, use the graph or the table to determine asolutio...
 3.3.48: In Exercises 4750, use the graph or the table to determine asolutio...
 3.3.49: In Exercises 4750, use the graph or the table to determine asolutio...
 3.3.50: In Exercises 4750, use the graph or the table to determine asolutio...
 3.3.51: a. Use synthetic division to show that 3 is a solution of thepolyno...
 3.3.52: a. Use synthetic division to show that 2 is a solution of thepolyno...
 3.3.53: In Exercises 5354, write a polynomial that represents the lengthof ...
 3.3.54: In Exercises 5354, write a polynomial that represents the lengthof ...
 3.3.55: During the 1980s, the controversial economist Arthur Lafferpromoted...
 3.3.56: During the 1980s, the controversial economist Arthur Lafferpromoted...
 3.3.57: Explain how to perform long division of polynomials. Use2x3  3x2 ...
 3.3.58: In your own words, state the Division Algorithm.
 3.3.59: How can the Division Algorithm be used to check the quotientand rem...
 3.3.60: Explain how to perform synthetic division. Use the divisionproblem ...
 3.3.61: State the Remainder Theorem.
 3.3.62: Explain how the Remainder Theorem can be used to findf(6) if f(x) ...
 3.3.63: How can the Factor Theorem be used to determine if x  1 isa factor...
 3.3.64: If you know that 2 is a zero off(x) = x3 + 7x2 + 4x  12,explain h...
 3.3.65: For each equation that you solved in Exercises 4346, use agraphing ...
 3.3.66: In Exercises 6669, determine whether eachstatement makes sense or d...
 3.3.67: In Exercises 6669, determine whether eachstatement makes sense or d...
 3.3.68: In Exercises 6669, determine whether eachstatement makes sense or d...
 3.3.69: In Exercises 6669, determine whether eachstatement makes sense or d...
 3.3.70: In Exercises 7073, determine whether each statement is true orfalse...
 3.3.71: In Exercises 7073, determine whether each statement is true orfalse...
 3.3.72: In Exercises 7073, determine whether each statement is true orfalse...
 3.3.73: In Exercises 7073, determine whether each statement is true orfalse...
 3.3.74: Find k so that 4x + 3 is a factor of20x3 + 23x2  10x + k.
 3.3.75: When 2x2  7x + 9 is divided by a polynomial, the quotient is2x  3...
 3.3.76: Find the quotient of x3n + 1 and xn + 1.
 3.3.77: Synthetic division is a process for dividing a polynomial byx  c. ...
 3.3.78: Use synthetic division to show that 5 is a solution ofx4  4x3  9x...
 3.3.79: Use the graph of y = f(x) to graph y = 2f(x + 1)  3.
 3.3.80: Given f(x) = 2x  3 and g(x) = 2x2  x + 5, find each ofthe followi...
 3.3.81: Find the inverse of f(x) = x  10x + 10.(Section 2.7, Example 4)
 3.3.82: Exercises 8284 will help you prepare for the material covered inthe...
 3.3.83: Exercises 8284 will help you prepare for the material covered inthe...
 3.3.84: Exercises 8284 will help you prepare for the material covered inthe...
Solutions for Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780134469164. This textbook survival guide was created for the textbook: College Algebra , edition: 7. Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems includes 84 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 84 problems in chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems have been answered, more than 29819 students have viewed full stepbystep solutions from this chapter.

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

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.

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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