 67.1: f(x) = x3  2x2  x + 1
 67.2: f(x) = 5x4  6x2 + 2
 67.3: Use synthetic substitution to estimate the sales for 2008.
 67.4: Use direct substitution to evaluate S(8).
 67.5: Which methodsynthetic substitution or direct substitutiondo you pre...
 67.6: x3  x2  5x  3; x + 1
 67.7: x3  3x + 2; x  1
 67.8: 6x3  25x2 + 2x + 8; 3x  2
 67.9: x4 + 2x3  8x  16; x + 2
 67.10: g(x) = x2  8x + 6
 67.11: g(x) = x3 + 2x2  3x + 1
 67.12: g(x) = x3  5x + 2
 67.13: g(x) = x4  6x  8
 67.14: g(x) = 2x3  8x2  2x + 5
 67.15: g(x) = 3x4 + x3  2x2 + x + 12
 67.16: g(x) = x5 + 8x3 + 2x  15
 67.17: g(x) = x6  4x4 + 3x2  10
 67.18: x3 + 2x2  x  2; x  1
 67.19: x3  x2  10x  8; x + 1
 67.20: x3 + x2  16x  16; x + 4
 67.21: x3  6x2 + 11x  6; x  2
 67.22: 2x3  5x2  28x + 15; x  5
 67.23: 3x3 + 10x2  x  12; x + 3
 67.24: 2x3 + 7x2  53x  28; 2x + 1
 67.25: 2x3 + 17x2 + 23x  42; 2x + 7
 67.26: x4 + 2x3 + 2x2  2x  3; x + 1
 67.27: 16x5  32x4  81x + 162; x  2
 67.28: Use synthetic substitution to show that x  8 is a factor of x3  4...
 67.29: Use the graph of the polynomial function at the right to determine ...
 67.30: Find the speed of the boat at 1, 2, and 3 seconds.
 67.31: It takes 6 seconds for the boat to travel between two buoys while i...
 67.32: Show that x  5 is a factor of the polynomial function.
 67.33: Are there other lengths of plastic that are extremely weak? Explain...
 67.34: (x2  x + k) (x  1) 3
 67.35: (x2 + kx  17) (x  2)
 67.36: (x2 + 5x + 7) (x + k) 3
 67.37: (x3 + 4x2 + x + k) (x + 2)
 67.38: Find his balance after 6 months if the annual interest rate is 12%....
 67.39: Find his balance after 6 months if the annual interest rate is 9.6%.
 67.40: How would the formula change if Zach wanted to pay the balance in f...
 67.41: Suppose he finances his purchase at 10.8% and plans to pay $410 eve...
 67.42: OPEN ENDED Give an example of a polynomial function that has a rema...
 67.43: REASONING Determine the dividend, divisor, quotient, and remainder ...
 67.44: CHALLENGE Consider the polynomial f(x) = ax4 + bx3 + cx2 + dx + e, ...
 67.45: Writing in Math Use the information on page 356 to explain how you ...
 67.46: ACT/SAT Use the graph of the polynomial function at the right. Whic...
 67.47: REVIEW The total area of a rectangle is 25a4  16b2. Which factors ...
 67.48: 7xy3  14x2y5 + 28x3y2
 67.49: ab  5a + 3b  15
 67.50: 2x2 + 15x + 25
 67.51: c3  216
 67.52: f(x) = x3  4x2 + x + 5
 67.53: f(x) = x4  6x3 + 10x2  x  3
 67.54: CITY PLANNING City planners have laid out streets on a coordinate g...
 67.55: x2 + 7x + 8 = 0
 67.56: 3x2  9x + 2 = 0
 67.57: 2x2 + 3x + 2 = 0
Solutions for Chapter 67: The Remainder and Factor Theorems
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 67: The Remainder and Factor Theorems
Get Full SolutionsChapter 67: The Remainder and Factor Theorems includes 57 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Since 57 problems in chapter 67: The Remainder and Factor Theorems have been answered, more than 122187 students have viewed full stepbystep solutions from this chapter.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.