 11.5.a: Find the second term in the expansion of
 11.5.1: The coefficients of a binomial expansion are called ________ ________.
 11.5.b: Find the fourth term in the expansion of 6C4 1 2 x 2 7y4 9003.75x 2...
 11.5.2: To find binomial coefficients, you can use the ________ ________ or...
 11.5.3: The notation used to denote a binomial coefficient is ________ or _...
 11.5.4: When you write out the coefficients for a binomial that is raised t...
 11.5.5:
 11.5.6:
 11.5.7:
 11.5.8:
 11.5.9:
 11.5.10:
 11.5.11: 10 4
 11.5.12: 10 6
 11.5.13: 100 98
 11.5.14: 100 2 1
 11.5.15: 6 5
 11.5.16: 9 6
 11.5.17: 7C4
 11.5.18: 10C2
 11.5.19: x 14
 11.5.20: x 16
 11.5.21: a 64
 11.5.22: a 55
 11.5.23: y 43
 11.5.24: y 25
 11.5.25: x y5
 11.5.26: c d3
 11.5.27: 2x y3
 11.5.28: 7a b3
 11.5.29: r 3s6
 11.5.30: r 3s6
 11.5.31: 3a 4b5
 11.5.32: 2x 5y5
 11.5.33: x 2 y24
 11.5.34: x 2 y 26
 11.5.35: 1 x y 5
 11.5.36: 1 x 2y 6
 11.5.37: 2 x y 4
 11.5.38: 2 x 3y 5
 11.5.39: 2 x 34 5 x 3 2
 11.5.40: 4x 13 2 4x 14 2
 11.5.41: 2t s5
 11.5.42: 3 2z4
 11.5.43: x 2y5
 11.5.44: 3v 26
 11.5.45: x y n 4
 11.5.46: x y n 7 6
 11.5.47: x 6y n 3 ,
 11.5.48: x 10z n 4 7
 11.5.49: 4x 3y n 8
 11.5.50: 5a 6b n 5
 11.5.51: 10x 3y n 10 ,
 11.5.52: 7x 2y n 7
 11.5.53: x 312
 11.5.54: x 2 312
 11.5.55: 4x y10 a
 11.5.56: x 2y10
 11.5.57: 2x 5y9 a
 11.5.58: 3x 4y8
 11.5.59: x 2 y10
 11.5.60: z 2 t10
 11.5.61: x 5 3
 11.5.62: 2t 1 3
 11.5.63: x 2 3 y1 33
 11.5.64: u3 5 25
 11.5.65: 3t 4 t 4
 11.5.66: x3 4 2x5 44
 11.5.67: f x x3
 11.5.68: f x x4
 11.5.69: f x x6
 11.5.70: f x x8
 11.5.71: f x x
 11.5.72: f x 1 x
 11.5.73:
 11.5.74:
 11.5.75: 2 3i6
 11.5.76: 5 9 3 2
 11.5.77: 1 2 3 2 i 3
 11.5.78: 5 3i 4
 11.5.79: 1.028
 11.5.80: 2.00510
 11.5.81: 2.9912
 11.5.82: 1.989
 11.5.83: f x x g x f x 4
 11.5.84: f x x g x f x 3 4
 11.5.85: A fair coin is tossed seven times. To find the probability of obtai...
 11.5.86: The probability of a baseball player getting a hit during any given...
 11.5.87: The probability of a sales representative making a sale with any on...
 11.5.88: To find the probability that the sales representative in Exercise 8...
 11.5.89: FINDING A PATTERN Describe the pattern formed by the sums of the nu...
 11.5.90: FINDING A PATTERN Use each of the encircledgroups of numbers in the...
 11.5.91: CHILD SUPPORT The average dollar amounts of child support collected...
 11.5.92: DATA ANALYSIS: ELECTRICITY The table shows the average prices (in c...
 11.5.93: The Binomial Theorem could be used to produce each row of Pascals T...
 11.5.94: A binomial that represents a difference cannot always be accurately...
 11.5.95: The term and the term of the expansion of have identical coeffici...
 11.5.96: WRITING In your own words, explain how to form the rows of Pascals ...
 11.5.97: Form rows 810 of Pascals Triangle.
 11.5.98: THINK ABOUT IT How many terms are in the expansion
 11.5.99: GRAPHICAL REASONING Which two functions have identical graphs, and ...
 11.5.100: CAPSTONE How do the expansions of and differ? Support your explanat...
 11.5.101: nCr nCn r
 11.5.102: nC0 nC1 nC2 . . . nCn 0 n
 11.5.103: n 1Cr nCr nCr 1
 11.5.104: The sum of the numbers in the th row of Pascals Triangle is
 11.5.105: Complete the table and describe the result. What characteristic of ...
 11.5.106: Another form of the Binomial Theorem is Use this form of the Binomi...
Solutions for Chapter 11.5: The Binomial Theorem
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 11.5: The Binomial Theorem
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Chapter 11.5: The Binomial Theorem includes 108 full stepbystep solutions. Since 108 problems in chapter 11.5: The Binomial Theorem have been answered, more than 52026 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.