Chapter 12.5, Problem 34AYU

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

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

To determine

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

Answer to Problem 34AYU

Solution:

The coefficient of $x^2$ in the expansion is $-314928$.

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,

x2 is $\left(\begin{array}{c}9\\ 9-2\end{array}\right){\left(-3\right)}^{9-2}{\left(2x\right)}^2 = \left(\begin{array}{c}9\\ 7\end{array}\right){\left(-3\right)}^7\mathrm{.4.}{x}^2 = 36\times -2187\times 4x^2 = -314928x^2$

The coefficient of $x^2$ in the expansion is $-314928$.