- 9.5.1: is a triangular display of the binomial coefficients.
- 9.5.2: a b = n 0 b = and . an1 a b =
- 9.5.3: True or False anjb = j!1n - j2! n!
- 9.5.4: The can be used to expand expressions like 12x + 326 .
- 9.5.5: a 5 3 b
- 9.5.6: a 7 3 b
- 9.5.7: a 7 5 b
- 9.5.8: a 9 7 b
- 9.5.9: a 50 49 b
- 9.5.10: a 100 98 b
- 9.5.11: a 1000 1000 b
- 9.5.12: a 1000 0 b
- 9.5.13: a 55 23 b
- 9.5.14: a 60 20b
- 9.5.15: a 47 25 b
- 9.5.16: a 37 19 b
- 9.5.17: 1x + 125
- 9.5.18: 1x - 125
- 9.5.19: 1x - 226
- 9.5.20: 1x + 325
- 9.5.21: 13x + 124
- 9.5.22: 12x + 325
- 9.5.23: 1x2 + y2 2 5
- 9.5.24: 1x2 - y2 2 6
- 9.5.25: A 1x + 22B 6
- 9.5.26: A 1x - 23B 4
- 9.5.27: 1ax + by25
- 9.5.28: 1ax - by24
- 9.5.29: The coefficient of in the expansion of 1x + 3210
- 9.5.30: The coefficient of in the expansion of 1x - 3210
- 9.5.31: The coefficient of in the expansion of 12x - 1212
- 9.5.32: The coefficient of in the expansion of 12x + 1212
- 9.5.33: The coefficient of in the expansion of 12x + 32
- 9.5.34: The coefficient of in the expansion of 12x - 329
- 9.5.35: The fifth term in the expansion of 1x + 327
- 9.5.36: The third term in the expansion of 1x - 327
- 9.5.37: The third term in the expansion of 13x - 229
- 9.5.38: The sixth term in the expansion of 13x + 228
- 9.5.39: The coefficient of in the expansion of ax2 + 1 x b 12
- 9.5.40: The coefficient of in the expansion of x - 1 x2 9
- 9.5.41: The coefficient of in the expansion of ax - 2 1x b 10
- 9.5.42: The coefficient of in the expansion of a 1x + 3 1x b 8
- 9.5.43: Use the Binomial Theorem to find the numerical value of correct to ...
- 9.5.44: Use the Binomial Theorem to find the numerical value of correct to ...
- 9.5.45: Show that and ann a b = 1.
- 9.5.46: Show that if n and j are integers with then anjb = a nn - jb Conclu...
- 9.5.47: f n is a positive integer, show that [Hint: 2 now use the Binomial ...
- 9.5.48: If n is a positive integer, show that an0b - an1b + an2b - + 1-12n ...
- 9.5.49: a 5 0 b a 1 4 b 5 + a 5 1 b a 1 4 b 4 a 3 4 b + a 5 2 b a 1 4 b 3 a...
- 9.5.50: Stirlings Formula An approximation for n!, when n is large, is give...
Solutions for Chapter 9.5: The Binomial Theorem
Full solutions for College Algebra | 9th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
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.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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 matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.