Chapter 12.5, Problem 38AYU

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

The 6th term in the expansion of ${\left(3x+2\right)}^8$

To determine

To find: The 6th term expansion of the given binomial expression ${\left(3x + 2\right)}^8$ using Binomial Theorem.

Answer to Problem 38AYU

Solution:

The 6th term in the expansion is $48384x^3.$

Explanation of Solution

Given:

Expression is given as ${\left(3x + 2\right)}^8$.

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 $n = 8$. Apply Binomial theorem till it reached 6th term.

${\left(3x + 2\right)}^8 = \left(\begin{array}{c}8\\ 0\end{array}\right){\left(3x\right)}^8 + \left(\begin{array}{c}8\\ 1\end{array}\right)\left(2\right){\left(3x\right)}^7 + \left(\begin{array}{c}8\\ 2\end{array}\right){\left(2\right)}^2{\left(3x\right)}^6 + \left(\begin{array}{c}8\\ 3\end{array}\right){\left(2\right)}^3{\left(3x\right)}^5 + \left(\begin{array}{c}8\\ 4\end{array}\right){\left(2\right)}^4{\left(3x\right)}^4 + \left(\begin{array}{c}8\\ 5\end{array}\right){\left(2\right)}^5{\left(3x\right)}^3 + \cdots$

The sixth term is

$\left(\begin{array}{c}8\\ 5\end{array}\right){\left(2\right)}^5{\left(3x\right)}^3 = 56 \times 32 \times 27.x^3 = 48384x^3$