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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. 234 (page 1214)

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial:- $a^{4} b^{2}$ term of ${(2 a 鈥� + 鈥� b)}^{6}$ .

Short Answer

Expert verified

The coefficient of $a^{4} b^{2}$ is 240 .

Step by step solution

01

Step 1. Given Information.

Given Binomial expression is ${(2 a 鈥� + 鈥� b)}^{6}$ .

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:-

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

Simplify the expression:-

$64 a^{6} + 192 a^{5} b + 240 a^{4} b^{2} + 160 a^{3} b^{3} + 60 a^{2} b^{4} + 12 a b^{5} + b^{6}$

Therefore, The coefficient of $a^{4} b^{2}$ is 240 .

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