Chapter 12.5, Problem 33AYU

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.

The coefficient of $x^7$ in the expansion of ${\left(2x+3\right)}^9$

To determine

To find: The coefficient of $x^7$ in the expansion of ${\left(2x + 3\right)}^9$ using Binomial Theorem.

Answer to Problem 33AYU

Solution:

The coefficient of $x^7$ in the expansion is 41472.

Explanation of Solution

Given:

Expression is given as ${\left(2x + 3\right)}^9$

Formula used:

The Binomial Theorem:

Let $x$ and $a$ be real numbers. For any positive integer $n$, we have

${\left(x + a\right)}^n = \left(\begin{array}{c}n\\ 0\end{array}\right)x^n + \left(\begin{array}{c}n\\ 1\end{array}\right)a{x}^{n-1} + \cdots + \left(\begin{array}{c}n\\ j\end{array}\right)a^j{x}^{n-j} + \cdots + \left(\begin{array}{c}n\\ n\end{array}\right)a^n$

${\left(x + a\right)}^n = {\displaystyle \sum _{k = 0}^n\left(\begin{array}{c}n\\ k\end{array}\right)x^{n-j}{a}^j}$

The binomial theorem can be used to find a particular term in an expansion without writing the entire expansion.

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

Here $x = 2x,a = 3$ and $n = 9$. Applying Binomial theorem,

$x^7$ is $\left(\begin{array}{c}9\\ 9-7\end{array}\right){\left(3\right)}^{9-7}{\left(2x\right)}^7 = \left(\begin{array}{c}9\\ 2\end{array}\right){\left(3\right)}^2\mathrm{.128.}{x}^3 = 36\times 9\times 128x^8 = 41472x^7$

The coefficient of $x^7$ in the expansion is 41472.