﻿ The Binomial Theorem Worksheet | Problems & Solutions

# The Binomial Theorem Worksheet

The Binomial Theorem Worksheet
• Page 1
1.
Find the coefficient of the term $x$8 in the expansion of ($x$ - 4)10.
 a. 740 b. 680 c. 700 d. 720

#### Solution:

The only term in the expansion (x - 4)10 that we need is 10C8 x8 (- 4)2.

= 10! / 8!2! x8 · 16

= 720 x8

Hence, the coefficient of x8 is 720.

2.
Find the coefficient of the term $x$7 in the expansion of ($x$ - 3)12.
 a. - 192465 b. - 192546 c. - 192654 d. - 192456

#### Solution:

The only term in the expansion (x - 3)12 that we need is 12C7 x7 (-3)5.

= - 12! / 7!5! · (243) · x7

= (- 192456) · x7

Hence, the coefficient of x7 is -192456.

3.
Find the coefficient of the term $x$10$y$4 in the binomial expansion ($x$ + $y$)14.
 a. 1002 b. 1000 c. 1004 d. 1001

#### Solution:

The only term in the expansion (x + y)14 that we need is 14C10 x10 y4.

= 14! / 10!4! x10 y4

= 1001 · x10y4

Hence, the coefficient of x10y4 is 1001.

4.
Find the coefficient of the term $x$5$y$6 in the expansion of ($x$ - 2$y$)11.
 a. 29568 b. 29586 c. 29865 d. 29658

#### Solution:

The only term in the expansion (x - 2y)11 that we need is 11C5 x5(-2y)6.

= 11! / 5!6! 64 x5y6

= 29568 x5y6

Hence, the coefficient of x5y6 is 29568.

5.
Find the coefficient of the term $x$3$y$3 the expansion of ($x$ - 8$y$)6.
 a. -40240 b. -10240 c. -30240 d. -20240

#### Solution:

The only term in the expansion (x - 8x)6 that we need is 6C3 x3(-8y)3.

= - 6! / 3!3! (512) x3y3

= (- 10240) x3y3

Hence, the coefficient of x3y3 is -10240.

6.
State the number of terms in the expansion of ($\mathrm{x + y}$)6 and give the first and last terms.
 a. 6, $x$7, $y$7 b. 6, $x$6, $y$6 c. 7, $x$6, $y$6 d. 7, $x$6$y$, $\mathrm{xy}$6

#### Solution:

First term: x 6
Last term: y 6
Number of terms: 6 + 1 = 7

7.
Find the expansion of the binomial function .
 a. ${a}^{5}+10{a}^{4}b+10{a}^{3}+5{a}^{2}$ b. ($\mathrm{a+b}$)($\mathrm{a+b}$)($\mathrm{a+b}$)($\mathrm{a+b}$)($\mathrm{a+b}$) c. ${a}^{5}+{b}^{5}$ d. ${a}^{5}+5{a}^{4}b+10{a}^{3}{b}^{2}+10{a}^{2}{b}^{3}+5a{b}^{4}+{b}^{5}$

#### Solution:

a5        a4b        a3b2        a2b3        ab4        b5
[Write the variable parts.]

1           5            10           10          5         1
[Find the co-efficients from PascalÃ¢â‚¬â„¢s triangle.]

Multiply the variable part of each term by its corresponding coefficient.

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

8.
Expand the binomial function: (2$\mathrm{x + y}$)4

#### Solution:

(2x + y)4 = [(2x) + (y)]4
[Write in (a + b)n form.]

(2x)4        (2x)3(y)        (2x)2y2        (2x)y3        y4
[ Write the variable parts. ]

16x4           8x3y          4x2y2            2xy3         y4
[Simplify.]

1                 4                   6                4            1
[Find the coefficients.]

Multiply the variable part of each term by its corresponding coefficient.

(2x + y)4 = 1(16x4) + 4(8x3y) + 6(4x2y2) + 4(2xy3) + 1(y4)

(2x + y)4 = 16x4 + 32x3y + 24x2y2 + 8xy3 + y4
[Simplify.]

9.
Expand:

#### Solution:

(5x + 3y)4 = [(5x) + (3y)]4
[Write in (a + b)n form.]

(5x)4        (5x)3(3y)        (5x)2(3y)2        (5x)(3y)3        (3y)4
[Write the variable parts.]

625x4        375x3y         225x2y2          135xy3           81y4
[Simplify.]

1                  4                        6                  4                    1
[Find the coefficients from PascalÃ¢â‚¬â„¢s triangle.]

To write the expansion, multiply the variable part of each term by its corresponding coefficient.

(5x + 3y)4 = 1(625x4) + 4(375x3y) + 6(225x2y2) + 4(135xy3) + 1(81y4)

(5x + 3y)4 = 625x4 + 1500x3y + 1350x2y2 + 540xy3 + 81y4

10.
Expand:

#### Solution:

(2x - 1)5 = [(2x) + (-1)]5
[Write the variable parts.]

(2x)5        (2x)4(-1)        (2x)3(-1)2        (2x)2(-1)3        (2x)(-1)4        (-1)5
[Write the variable parts.]

32x5           -16x4              8x3                 - 4x2                     2x            -1
[Simplify.]

1                 5                    10                    10                       5                 1
[Find the coefficients from PascalÃ¢â‚¬â„¢s triangle.]

To write the expansion, multiply the variable part of each term by its corresponding coefficient.

(2x - 1)5 = 1(32x5) + 5(-16x4) + 10(8x³) +10(- 4x²) + 5(2x) + 1(-1)

(2x - 1)5 = 32x5 - 80x4 + 80x³ - 40x² + 10x - 1