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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. 220 (page 1213)

In the following exercises, expand each binomial ${(p 鈥� 鈭� 鈥� 1)}^{5}$ .

Short Answer

Expert verified

The solution of the exercise is $p^{5} - 5 p^{4} + 10 p^{3} - 10 p^{2} + 5 p - 1$ .

Step by step solution

01

Step 1. Given Information.

Given Binomial expression is ${(p 鈥� 鈭� 鈥� 1)}^{5}$ .

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:-

${(p 鈥� 鈭� 鈥� 1)}^{5} = {鈭�}_{i = 0}^{5} (\binom{5}{i}) p^{(5 - i)} {(- 1)}^{i}$

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

Simplify the expression:-

$p^{5} - 5 p^{4} + 10 p^{3} - 10 p^{2} + 5 p - 1$

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