 6.7.1: Use synthetic substitution to find f(3) and f(4) for each function.
 6.7.2: Use synthetic substitution to find f(3) and f(4) for each function.
 6.7.3: Use synthetic substitution to estimate the sales for 2008
 6.7.4: Use direct substitution to evaluate S(8)
 6.7.5: Which methodsynthetic substitution or direct substitutiondo you pre...
 6.7.6: Given a polynomial and one of its factors, find the remaining facto...
 6.7.7: Given a polynomial and one of its factors, find the remaining facto...
 6.7.8: Given a polynomial and one of its factors, find the remaining facto...
 6.7.9: Given a polynomial and one of its factors, find the remaining facto...
 6.7.10: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.11: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.12: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.13: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.14: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.15: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.16: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.17: Use synthetic substitution to find g(3) and g(4) for each function
 6.7.18: Given a polynomial and one of its factors, find the remaining facto...
 6.7.19: Given a polynomial and one of its factors, find the remaining facto...
 6.7.20: Given a polynomial and one of its factors, find the remaining facto...
 6.7.21: Given a polynomial and one of its factors, find the remaining facto...
 6.7.22: Given a polynomial and one of its factors, find the remaining facto...
 6.7.23: Given a polynomial and one of its factors, find the remaining facto...
 6.7.24: Given a polynomial and one of its factors, find the remaining facto...
 6.7.25: Given a polynomial and one of its factors, find the remaining facto...
 6.7.26: Given a polynomial and one of its factors, find the remaining facto...
 6.7.27: Given a polynomial and one of its factors, find the remaining facto...
 6.7.28: Use synthetic substitution to show that x  8 is a factor of x3  4...
 6.7.29: Use the graph of the polynomial function at the right to determine ...
 6.7.30: Find the speed of the boat at 1, 2, and 3 seconds.
 6.7.31: It takes 6 seconds for the boat to travel between two buoys while i...
 6.7.32: Show that x  5 is a factor of the polynomial function.
 6.7.33: Are there other lengths of plastic that are extremely weak? Explain...
 6.7.34: Find values of k so that each remainder is 3.
 6.7.35: Find values of k so that each remainder is 3.
 6.7.36: Find values of k so that each remainder is 3.
 6.7.37: Find values of k so that each remainder is 3.
 6.7.38: Find his balance after 6 months if the annual interest rate is 12%....
 6.7.39: Find his balance after 6 months if the annual interest rate is 9.6%.
 6.7.40: How would the formula change if Zach wanted to pay the balance in f...
 6.7.41: Suppose he finances his purchase at 10.8% and plans to pay $410 eve...
 6.7.42: Give an example of a polynomial function that has a remainder of 5 ...
 6.7.43: Determine the dividend, divisor, quotient, and remainder represente...
 6.7.44: Consider the polynomial f(x) = ax4 + bx3 + cx2 + dx + e, where a + ...
 6.7.45: Use the information on page 356 to explain how you can use the Rema...
 6.7.46: Use the graph of the polynomial function at the right. Which is not...
 6.7.47: The total area of a rectangle is 25a4  16b2. Which factors could r...
 6.7.48: Factor completely. If the polynomial is not factorable, write prime.
 6.7.49: Factor completely. If the polynomial is not factorable, write prime.
 6.7.50: Factor completely. If the polynomial is not factorable, write prime.
 6.7.51: Factor completely. If the polynomial is not factorable, write prime.
 6.7.52: Graph each function by making a table of values
 6.7.53: Graph each function by making a table of values
 6.7.54: City planners have laid out streets on a coordinate grid before beg...
 6.7.55: Find the exact solutions of each equation by using the Quadratic Fo...
 6.7.56: Find the exact solutions of each equation by using the Quadratic Fo...
 6.7.57: Find the exact solutions of each equation by using the Quadratic Fo...
Solutions for Chapter 6.7: The Remainder and Factor Theorems
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 6.7: The Remainder and Factor Theorems
Get Full SolutionsChapter 6.7: The Remainder and Factor Theorems includes 57 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. Since 57 problems in chapter 6.7: The Remainder and Factor Theorems have been answered, more than 47650 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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!).

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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

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).

Outer product uv T
= column times row = rank one matrix.

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.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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.