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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. 230 (page 1214)

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial :- $y^{3}$ term of ${(y 鈥� + 鈥� 5)}^{4}$ .

Short Answer

Expert verified

The coefficient of $y^{3}$ for the binomial is 20 .Step 1. Given Information.

Step by step solution

01

Step 1. Given Information.

The binomial expression is ${(y 鈥� + 鈥� 5)}^{4}$ .

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:-

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

Simplify the expression:-

$y^{4} + 20 y^{3} + 150 y^{2} + 500 y + 625$

Therefore, The coefficient of $y^{3}$ is 20 .

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Most popular questions from this chapter

In the following exercises, expand each binomial using Pascal's Triangle.

$(2 x + 3 y)^{3}$

In the following exercises, evaluate.

(a) $(\begin{array}{l} 3 \\ 1 \end{array})$ (b) $(\begin{array}{l} 9 \\ 9 \end{array})$ (c) $(\begin{array}{l} 7 \\ 0 \end{array})$ (d) $(\begin{array}{l} 5 \\ 3 \end{array})$

Determine if each sequence is arithmetic. If so, indicate the common difference.

$\begin{array}{l} (a) 9, 20, 31, 42, 53, 64, 鈥� \\ (b) 12, 6, 0, - 6, - 12, - 18, 鈥� \\ (c) 7, 1, 10, 4, 13, 7, 鈥� \end{array}$

Write the first five terms of the sequence whose general term is $a_{n} = \frac{2}{n!} .$

Expand the partial sum and find its value: ${鈭�}_{k = 0}^{3} \frac{2}{k!}$

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