 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  + 112n ...
 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
ISBN: 9780321716811
Solutions for Chapter 9.5: The Binomial Theorem
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321716811. Since 50 problems in chapter 9.5: The Binomial Theorem have been answered, more than 8045 students have viewed full stepbystep solutions from this chapter. Chapter 9.5: The Binomial Theorem includes 50 full stepbystep solutions.

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).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Kirchhoff's Laws.
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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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 matrix A.
A square matrix that has no inverse: det(A) = o.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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