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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. 222 (page 1213)

In the following exercises, expand each binomial ${(3 x 鈥� 鈭� 鈥� y)}^{5}$ .

Short Answer

Expert verified

The solution of the exercise is $243 x^{5} - 405 x^{4} y + 270 x^{3} y^{2} - 90 x^{2} y^{3} + 15 x y^{4} - y^{5}$ .

Step by step solution

01

Step 1. Given Information.

Given Binomial expression is ${(3 x 鈥� 鈭� 鈥� y)}^{5}$ .

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:-

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

Simplify the expression:-

$243 x^{5} - 405 x^{4} y + 270 x^{3} y^{2} - 90 x^{2} y^{3} + 15 x y^{4} - y^{5}$

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Most popular questions from this chapter

In the following exercises, evaluate.
$(a) (\begin{array}{l} 7 \\ 1 \end{array}) (b) (\begin{array}{l} 4 \\ 4 \end{array}) (c) (\begin{array}{l} 3 \\ 0 \end{array}) (d) (\begin{array}{l} 10 \\ 8 \end{array})$

Find a general term for the sequence whose first five terms are shown 1, 鈭�4, 9, 鈭�16, 25,...

Find the sum of the first $20$ terms of the geometric sequence $3, 6, 12, 24, 48, 96, 鈥�$ .

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, . . . .$

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

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