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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 29. (page 836)

The coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10}$

Short Answer

Expert verified

The coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10}$ is $17010$ .

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 $x^{6}$ in the expansion of ${(x + 3)}^{10}$ is:

localid="1647190158816" $= (\begin{matrix} 10 \\ 10 - 6 \end{matrix}) {(3)}^{10 - 6} x^{6}$

$= \frac{10!}{4! (10 - 4)!} {(3)}^{4} x^{6}$

$= \frac{10 脳 9 脳 8 脳 7 脳 6!}{4! 脳 6!} (81) x^{6}$

$= (210) (81) x^{6}$

$= 17010 x^{6}$

So, the coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10} = 17010$ .

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