Chapter 12.5, Problem 37AYU

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

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

To determine

To find: The 3rd term expansion of the given binomial expression ${\left(3x - 2\right)}^9$ using Binomial Theorem.

Answer to Problem 37AYU

Solution:

The 3rd term in the expansion is $314928x^7$.

Explanation of Solution

Given:

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

Formula used:

The Binomial Theorem:

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

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

Here $x = 3x,a = - 2$ and $n = 9$. Apply Binomial theorem till it reached 3rd term

${\left(3x - 2\right)}^9 = \left(\begin{array}{c}9\\ 0\end{array}\right){\left(3x\right)}^9 + \left(\begin{array}{c}9\\ 1\end{array}\right)\left( - 2\right){\left(3x\right)}^8 + \left(\begin{array}{c}9\\ 2\end{array}\right){\left( - 2\right)}^2{\left(3x\right)}^7 + \cdots$

The third term is

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