 3.3.1: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.2: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.3: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.4: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.5: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.6: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.7: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.8: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.9: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.10: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.11: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.12: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.13: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.14: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.15: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.16: In Exercises 116. di11ide using long division. State tht! quo#ent,...
 3.3.17: In Exercises 17 32, divide using synthetic division. (2t2 + x  10...
 3.3.18: In Exercises 17 32, divide using synthetic division. (x2 + x  2) ...
 3.3.19: In Exercises 17 32, divide using synthetic division. 3x2 + 1x  20...
 3.3.20: In Exercises 17 32, divide using synthetic division. (5x2  12t  ...
 3.3.21: In Exercises 17 32, divide using synthetic division. (4x3  3.,.2 ...
 3.3.22: In Exercises 17 32, divide using synthetic division. (5x3  6x2 + ...
 3.3.23: In Exercises 17 32, divide using synthetic division. (6x5  2x' + ...
 3.3.24: In Exercises 17 32, divide using synthetic division. (x' + 4x'  3...
 3.3.25: In Exercises 17 32, divide using synthetic division. (x2  5x  5x...
 3.3.26: In Exercises 17 32, divide using synthetic division. (x2  6x  6x...
 3.3.27: In Exercises 17 32, divide using synthetic division. x5 + x3  2 X...
 3.3.28: In Exercises 17 32, divide using synthetic division. x1 +x'  lOx'...
 3.3.29: In Exercises 17 32, divide using synthetic division. X~ ~6
 3.3.30: In Exercises 17 32, divide using synthetic division. li8
 3.3.31: In Exercises 17 32, divide using synthetic division. r.s  3x4 + x...
 3.3.32: In Exercises 17 32, divide using synthetic division. r5  b.4  r...
 3.3.33: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.34: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.35: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.36: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.37: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.38: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.39: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.40: In Exercises 3340, use synthetic division and the Remainder Theore...
 3.3.41: Use synthetic division to divide f(x)  x'  4x 2 + x + 6byx+ I. Us...
 3.3.42: Use synthetic di\'ision to divide f(x)  x'  2r'  x + 2 by x ~ I....
 3.3.43: Solve the equation 2t3  5x2 + x + 2  0 given lhat 2 is a zero of ...
 3.3.44: Solve the equation 2..2  3x2  llx + 6  0 given that  2 is a ze...
 3.3.45: Solve the equation 12.J + 16x2  5x  3  0 given that ~ is a root.
 3.3.46: Solve the equation 3x3 + 7x2  22r  8  0 given that  t is a root.
 3.3.47: In Exercises 4750, use the graph or the table to determine a solut...
 3.3.48: In Exercises 4750, use the graph or the table to determine a solut...
 3.3.49: In Exercises 4750, use the graph or the table to determine a solut...
 3.3.50: In Exercises 4750, use the graph or the table to determine a solut...
 3.3.51: a. Use synthetic division to show that 3 is a solution of the polyn...
 3.3.52: a. Use synthetic division to show that 2 is a solution of the JX>Iy...
 3.3.53: In Exercises 5354, write a polynomial that represents the length o...
 3.3.54: In Exercises 5354, write a polynomial that represents the length o...
 3.3.55: During the 980.l~ the cotJtroJ>'ersial economist Arthur Laffer prom...
 3.3.56: During the 980.l~ the cotJtroJ>'ersial economist Arthur Laffer prom...
 3.3.57: Explain how to perform long division of polynomials. Use 2t3  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 quotient and re...
 3.3.60: Explain how to perform synthetic division. Use the division problem...
 3.3.61: State the Remainder Theorem.
 3.3.62: Explain how the Remainder Theorem can be tL'\ed to find !( 6) if !...
 3.3.63: How can the Facto r Theorem be used to dete rmine if x  1 is a fac...
 3.3.64: If you know tha t  2 is a 1.eroof j{x)  x3 + 7x2 + 4x  12, expla...
 3.3.65: For each equation that you solved in Exercises 4346, use a graphin...
 3.3.66: In Exercises 6669, detemine wlzeJher each srafemem makes SI!JISe o...
 3.3.67: In Exercises 6669, detemine wlzeJher each srafemem makes SI!JISe o...
 3.3.68: In Exercises 6669, detemine wlzeJher each srafemem makes SI!JISe o...
 3.3.69: In Exercises 6669, detemine wlzeJher each srafemem makes SI!JISe o...
 3.3.70: In Exercises 7073, determine whether each statement is true or fal...
 3.3.71: In Exercises 7073, determine whether each statement is true or fal...
 3.3.72: In Exercises 7073, determine whether each statement is true or fal...
 3.3.73: In Exercises 7073, determine whether each statement is true or fal...
 3.3.74: Find k so that 4x + 3 is a factor of 20x' + 23x1  lOx + k.
 3.3.75: When 2t2  7.t + 9 is divided by a polynomial, the quotient is 2t ...
 3.3.76: Find the quotient of x3 " + I and.<" + I.
 3.3.77: Synthetic division is a process for d ividing a polynomial by x  c...
 3.3.78: Use synthetic division to show that 5 is a solution of x'  4.<3  ...
 3.3.79: Exercises 7981 will help yoa prepare for 1/re material covered in ...
 3.3.80: Exercises 7981 will help yoa prepare for 1/re material covered in ...
 3.3.81: Exercises 7981 will help yoa prepare for 1/re material covered in ...
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
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 6. Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems includes 81 full stepbystep 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 stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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 = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

lAII = 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 = MI 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 AI 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 = UI.
Orthonormal columns (complex analog of Q).