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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q 30. (page 836)

The coefficient of $x^{3}$ in the expansion of ${(x - 3)}^{10}$

Short Answer

Expert verified

The coefficient of $x^{3}$ in the expansion of ${(x - 3)}^{10}$ is $- 262440$ .

Step by step solution

01

Step 1. Definition

Based on the expansion of ${(x + a)}^{n}$ , the term containing $x^{j}$ is $(\begin{matrix} n \\ n - j \end{matrix}) a^{n - j} x^{j}$ .

02

Step 2. Explanation

The term containing $x^{3}$ in the expansion of ${(x - 3)}^{10}$ is:

$= (\begin{matrix} 10 \\ 10 - 3 \end{matrix}) {(- 3)}^{10 - 3} x^{3}$

= \frac{10!}{7! (10 - 7)!} {(- 3)}^{7} x^{3}

= \frac{10 脳 9 脳 8 脳 7!}{7! 脳 3!} (- 2187) x^{3}

$= (120) (- 2187) x^{3}$

$= - 262440 x^{3}$

So, the coefficient of $x^{3}$ in the expansion of ${(x - 3)}^{10} = - 262440$ .

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