Download HOMEWORK #5 The Binomial Theorem

Pre-Calculus 12 – Ch. 11 Permutations and Combinations
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HOMEWORK #5 The Binomial Theorem
1) Use the Binomial Theorem to expand each of the following:
a) x 1
5
b) 2x  1
6
c) 2  x 
5
d) 2x  3y 
4
2) Use the general term of the Binomial Theorem to determine the indicated term:
a) the third term of x  7 
7
1

b) the fourth term of  2x  

3
8
c) the sixth term of 4  3x 
9
d) the eleventh term of 2x  y 
13
3) What is the greatest numerical coefficient in the expansion of 3x 1 ?
8
4) Factor each in the form a  b  .
n
a) k 3  3k 2 m  3km2  m 3
b) x 5  5x 4 y 10x 3 y2 10x 2 y 3  5xy 4  y5
c) x 3  6x 2 12x  8
9
1

5) Find the term involving x of  x   and simplify.

2x 
7
6) Find the numerical coefficient of x 5 in
a) 2  x 
7
b) 1 3x 
5
SOLUTIONS
1)
a)
5
C0 x 5  5 C1x 4  5 C2 x 3  5 C3 x 2  5 C4 x  5 C5
 x 5  5x 4 10x 3 10x 2  5x 1
b)
C0 2x   6 C1 2x   6 C2 2x   6 C3 2x   6 C4 2x   6 C5 2x   6 C6 (2x)0
6
6
5
4
3
2
1
 64x 6 192x 5  240x 4 160x 3  60x 2 12x 1
c)
5
C0 (2)5 (x)0  5 C1 (2)4 (x)  5 C2 (2)3 (x)2  5 C3 (2)2 (x)3  5 C4 (2)(x)4  5 C5 (x)5
 32  80x  80x 2  40x 3 10x 4  x 5
d)
C0 2x  3y  4 C1 4x  3y  4 C2 4x  3y  4 C3 4x 3y  4 C4 3y
4
4
0
3
1
2
2
3
4
 16x 4  96x 3 y  216x 2 y 2  216xy 3  81y 4
2)
a)
b)
t k 1  n C k a n k b k
t k1  n Ck a nk b k
5 1
t 4  8 C 3 2x   
 3
1792 5
t4 
x
27
t 3  7 C2 x  7 
5
2
t 3  1029x 5
c)
d)
t k1  n Ck a
nk
b
t k1  n Ck a nk b k
k
t 6  9 C5 4  3x 
t11  13 C10 2x  y
t 6  7838208x 5
t11  2288x 3 y10
4
5
3
4) a) k  m 
3
3)
10
b) x  y 
5
C0 3x   8 C1 3x   8 C2 3x   8 C3 3x   ...
8
8
3
7
6
5
 6561x 8 17496x 7  20412x 6 13608x 5  ...
Greatest = 20412
5)
6) a)
b)
0
9 1 
t1  9 C0 x     x 9
 2x 
1
 1  9
t 2  9 C1 x     x 7
 2x  2
8
C5 2  x 
2
7
 214 
 84
5
C5 3x 
5
5
 243
c) x  2 
3