 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 34033 students have viewed full stepbystep solutions from this chapter.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Solvable system Ax = b.
The right side b is in the column space of A.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).