 14.4.14.1.341: Fill in each blank so that the resulting statement is true. n r is ...
 14.4.14.1.342: Fill in each blank so that the resulting statement is true. 8 2 ! ! !
 14.4.14.1.343: Fill in each blank so that the resulting statement is true. n r
 14.4.14.1.344: Fill in each blank so that the resulting statement is true. x 25 5 ...
 14.4.14.1.345: Fill in each blank so that the resulting statement is true. (a b)n ...
 14.4.14.1.346: Fill in each blank so that the resulting statement is true. The for...
 14.4.14.1.347: Fill in each blank so that the resulting statement is true. The (r ...
 14.4.14.1.348: In Exercises 18, evaluate the given binomial coefficient 8 3
 14.4.14.1.349: In Exercises 18, evaluate the given binomial coefficient 7 2
 14.4.14.1.350: In Exercises 18, evaluate the given binomial coefficient 12 1
 14.4.14.1.351: In Exercises 18, evaluate the given binomial coefficient 11 1
 14.4.14.1.352: In Exercises 18, evaluate the given binomial coefficient 6 6
 14.4.14.1.353: In Exercises 18, evaluate the given binomial coefficient 15 2
 14.4.14.1.354: In Exercises 18, evaluate the given binomial coefficient 100 2
 14.4.14.1.355: In Exercises 18, evaluate the given binomial coefficient 100 98
 14.4.14.1.356: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.357: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.358: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.359: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.360: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.361: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.362: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.363: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.364: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.365: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.366: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.367: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.368: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.369: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.370: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.371: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.372: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.373: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.374: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.375: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.376: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.377: In Exercises 930, use the Binomial Theorem to expand each binomial ...
 14.4.14.1.378: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.379: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.380: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.381: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.382: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.383: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.384: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.385: In Exercises 3138, write the first three terms in each binomial exp...
 14.4.14.1.386: In Exercises 3948, find the indicated term in each expansion. (2x y...
 14.4.14.1.387: In Exercises 3948, find the indicated term in each expansion. (x 2y...
 14.4.14.1.388: In Exercises 3948, find the indicated term in each expansion. (x 1)...
 14.4.14.1.389: In Exercises 3948, find the indicated term in each expansion. (x 1)...
 14.4.14.1.390: In Exercises 3948, find the indicated term in each expansion. (x2 y...
 14.4.14.1.391: In Exercises 3948, find the indicated term in each expansion. (x3 y...
 14.4.14.1.392: In Exercises 3948, find the indicated term in each expansion. x 12 ...
 14.4.14.1.393: In Exercises 3948, find the indicated term in each expansion. x 12 ...
 14.4.14.1.394: In Exercises 3948, find the indicated term in each expansion. (x2 y...
 14.4.14.1.395: In Exercises 3948, find the indicated term in each expansion. (x 2y...
 14.4.14.1.396: In Exercises 4952, use the Binomial Theorem to expand each expressi...
 14.4.14.1.397: In Exercises 4952, use the Binomial Theorem to expand each expressi...
 14.4.14.1.398: In Exercises 4952, use the Binomial Theorem to expand each expressi...
 14.4.14.1.399: In Exercises 4952, use the Binomial Theorem to expand each expressi...
 14.4.14.1.400: Exercises 5354 involve expressions containing i, where i 1. Expand ...
 14.4.14.1.401: Exercises 5354 involve expressions containing i, where i 1. Expand ...
 14.4.14.1.402: In Exercises 5556, find f (x h) f (x) h and simplify. f (x) x4 7
 14.4.14.1.403: In Exercises 5556, find f (x h) f (x) h and simplify. f (x) x5 8
 14.4.14.1.404: Find the middle term in the expansion of 3 x x 3 10 .
 14.4.14.1.405: Find the middle term in the expansion of 1 x x2 12 .
 14.4.14.1.406: The probability that a smoker suffers from depression is 0.28. If f...
 14.4.14.1.407: The probability that a person in the general population suffers fro...
 14.4.14.1.408: Explain how to evaluate n r . Provide an example with your explanat...
 14.4.14.1.409: Describe the pattern in the exponents on a in the expansion of (a b)n.
 14.4.14.1.410: Describe the pattern in the exponents on b in the expansion of (a b)n.
 14.4.14.1.411: What is true about the sum of the exponents on a and b in any term ...
 14.4.14.1.412: How do you determine how many terms there are in a binomial expansion?
 14.4.14.1.413: Explain how to use the Binomial Theorem to expand a binomial. Provi...
 14.4.14.1.414: Explain how to find a particular term in a binomial expansion witho...
 14.4.14.1.415: Use the nCr key on a graphing utility to verify your answers in Exe...
 14.4.14.1.416: In Exercises 6970, graph each of the functions in the same viewing ...
 14.4.14.1.417: In Exercises 6970, graph each of the functions in the same viewing ...
 14.4.14.1.418: In Exercises 7173, use the Binomial Theorem to find a polynomial ex...
 14.4.14.1.419: In Exercises 7173, use the Binomial Theorem to find a polynomial ex...
 14.4.14.1.420: In Exercises 7173, use the Binomial Theorem to find a polynomial ex...
 14.4.14.1.421: In order to expand (x3 y4)5, I find it helpful to rewrite the expre...
 14.4.14.1.422: Without writing the expansion of (x 1)6, I can see that the terms h...
 14.4.14.1.423: I use binomial coefficients to expand (a b)n, where n 1 is the coef...
 14.4.14.1.424: One of the terms in my binomial expansion is 7 5 x2y4.
 14.4.14.1.425: In Exercises 7881, determine whether each statement is true or fals...
 14.4.14.1.426: In Exercises 7881, determine whether each statement is true or fals...
 14.4.14.1.427: In Exercises 7881, determine whether each statement is true or fals...
 14.4.14.1.428: In Exercises 7881, determine whether each statement is true or fals...
 14.4.14.1.429: Use the Binomial Theorem to expand and then simplify the result: (x...
 14.4.14.1.430: Find the term in the expansion of (x2 y2)5 containing x4 as a factor.
 14.4.14.1.431: If f(x) x2 2x 3, find f(a 1). (Section 8.1, Example 3)
 14.4.14.1.432: If f(x) x2 5x and g(x) 2x 3, find f(g(x)) and g(f(x)). (Section 8.4...
 14.4.14.1.433: Subtract: x x 3 x 1 2x2 2x 24 . (Section 7.4, Example 7)
Solutions for Chapter 14.4: The Binomial Theorem
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 14.4: The Binomial Theorem
Get Full SolutionsChapter 14.4: The Binomial Theorem includes 93 full stepbystep solutions. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Since 93 problems in chapter 14.4: The Binomial Theorem have been answered, more than 71081 students have viewed full stepbystep solutions from this chapter.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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 .

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).

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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 ().

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.

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