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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. 232 (page 1214)

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial :- $x^{5}$ term of ${(x 鈥� 鈭� 鈥� 4)}^{6}$ .

Short Answer

Expert verified

The coefficient of $x^{5}$ is -24 .

Step by step solution

01

Step 1. Given Information.

Given Binomial expression is ${(x 鈥� 鈭� 鈥� 4)}^{6}$ .

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:-

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

Simplify the expression:-

$x^{6} - 24 x^{5} + 240 x^{4} - 1280 x^{3} + 3840 x^{2} - 6144 x + 4096$

Therefore, The coefficient of $x^{5}$ is -24 .

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Most popular questions from this chapter

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