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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 12: Q. 231 (page 1214)

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial :- $x^{6}$ term of ${(x 鈥� + 鈥� 2)}^{8}$ .

Short Answer

Expert verified

The coefficient of $x^{6}$ is 112 .

Step by step solution

01

Step 1. Given Information.

Given Binomial expression is ${(x 鈥� + 鈥� 2)}^{8}$ .

02

Step 2. Binomial expansion formula .

The binomial expansion formula is :-

${(a + b)}^{n} = {鈭�}_{i = 0}^{n} (\binom{n}{i}) a^{(n - i)} b^{i}$

03

Step 3. Solve Binomial Expansion

Expanding using Binomial theorem:-

${(x 鈥� + 鈥� 2)}^{8} = {鈭�}_{i = 0}^{8} (\binom{8}{i}) x^{(8 - i)} 路鈥� 2^{i} {鈭�}_{i = 0}^{8} (\binom{8}{i}) x^{(8 - i)} 路鈥� 2^{i} = \frac{8!}{0! (8 - 0)!} x^{8} 路鈥� 2^{0} + \frac{8!}{1! (8 - 1)!} x^{7} 路鈥� 2^{1} + \frac{8!}{2! (8 - 2)!} x^{6} 路鈥� 2^{2} + \frac{8!}{3! (8 - 3)!} x^{5} 路鈥� 2^{3} + \frac{8!}{4! (8 - 4)!} x^{4} 路鈥� 2^{4} + \frac{8!}{5! (8 - 5)!} x^{3} 路鈥� 2^{5} + \frac{8!}{6! (8 - 6)!} x^{2} 路鈥� 2^{6} + \frac{8!}{7! (8 - 7)!} x^{1} 路鈥� 2^{7} + \frac{8!}{8! (8 - 8)!} x^{0} 路鈥� 2^{8}$

Simplify the expression:-

$x^{8} + 16 x^{7} + 112 x^{6} + 448 x^{5} + 1120 x^{4} + 1792 x^{3} + 1792 x^{2} + 1024 x + 256$

Therefore, The coefficient of $x^{6}$ is 112 .

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