 8.5.73: In Exercises 7378, use the Binomial Theorem to expand thecomplex nu...
 8.5.74: In Exercises 7378, use the Binomial Theorem to expand thecomplex nu...
 8.5.75: In Exercises 7378, use the Binomial Theorem to expand thecomplex nu...
 8.5.76: In Exercises 7378, use the Binomial Theorem to expand thecomplex nu...
 8.5.77: In Exercises 7378, use the Binomial Theorem to expand thecomplex nu...
 8.5.78: In Exercises 7378, use the Binomial Theorem to expand thecomplex nu...
 8.5.79: APPROXIMATION In Exercises 7982, use the BinomialTheorem to approxi...
 8.5.80: APPROXIMATION In Exercises 7982, use the BinomialTheorem to approxi...
 8.5.81: APPROXIMATION In Exercises 7982, use the BinomialTheorem to approxi...
 8.5.82: APPROXIMATION In Exercises 7982, use the BinomialTheorem to approxi...
 8.5.83: GRAPHICAL REASONING In Exercises 83 and 84, use agraphing utility t...
 8.5.84: GRAPHICAL REASONING In Exercises 83 and 84, use agraphing utility t...
 8.5.85: PROBABILITY In Exercises 8588, consider independenttrials of an exp...
 8.5.86: PROBABILITY In Exercises 8588, consider independenttrials of an exp...
 8.5.87: PROBABILITY In Exercises 8588, consider independenttrials of an exp...
 8.5.88: To find the probability that the sales representative inExercise 87...
 8.5.89: FINDING A PATTERN Describe the pattern formedby the sums of the num...
 8.5.90: FINDING A PATTERN Use each of the encircledgroups of numbers in the...
 8.5.91: CHILD SUPPORT The average dollar amounts ofchild support collected ...
 8.5.92: DATA ANALYSIS: ELECTRICITY The table shows theaverage prices (in ce...
 8.5.93: TRUE OR FALSE? In Exercises 9395, determine whetherthe statement is...
 8.5.94: TRUE OR FALSE? In Exercises 9395, determine whetherthe statement is...
 8.5.95: TRUE OR FALSE? In Exercises 9395, determine whetherthe statement is...
 8.5.96: WRITING In your own words, explain how to formthe rows of Pascals T...
 8.5.97: Form rows 810 of Pascals Triangle.
 8.5.98: THINK ABOUT IT How many terms are in the expansion ofx yn?
 8.5.99: GRAPHICAL REASONING Which two functionshave identical graphs, and w...
 8.5.100: CAPSTONE How do the expansions ofand differ? Support your explanati...
 8.5.101: PROOF In Exercises 101104, prove the property for allintegers and w...
 8.5.102: PROOF In Exercises 101104, prove the property for allintegers and w...
 8.5.103: PROOF In Exercises 101104, prove the property for allintegers and w...
 8.5.104: PROOF In Exercises 101104, prove the property for allintegers and w...
 8.5.105: Complete the table and describe the result. What characteristic of ...
 8.5.106: Another form of the Binomial Theorem isUse this form of the Binomia...
Solutions for Chapter 8.5: The Binomial Theorem
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 8.5: The Binomial Theorem
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 8. College Algebra was written by and is associated to the ISBN: 9781439048696. Since 34 problems in chapter 8.5: The Binomial Theorem have been answered, more than 33445 students have viewed full stepbystep solutions from this chapter. Chapter 8.5: The Binomial Theorem includes 34 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.