Question

use the Binomial Theorem to find the indicated coefficient or term.

The third term in the expansion of (x - 3)⁷

Short answer

The third term is 324x⁵

Step-by-step solution

. Given information 

(x-3)⁷ is given

We have to find out the third term

. Expansion of  (x-3)7 using The Binomial Theorem 

the binomial theorem states that

$$\begin{gathered} Letxandaberealnumbers.Foranypositiveintegern,wehave \\ {(x+a)}^n=\left(\begin{array}{c}n\\ 0\end{array}\right)x^n+\left(\begin{array}{c}n\\ 1\end{array}\right)a{x}^{n-1}+\left(\begin{array}{c}n\\ 2\end{array}\right)a^2{x}^{n-2}+.......+\left(\begin{array}{c}n\\ j\end{array}\right)a^j{x}^{n-j}+.....+\left(\begin{array}{c}n\\ n\end{array}\right)a^n \end{gathered}$$

using this we can expand the given expression

localid="1647340126933" $$\begin{gathered} {(x-3)}^7=\left(\begin{array}{c}7\\ 0\end{array}\right)x^7+\left(\begin{array}{c}7\\ 1\end{array}\right)\times {(-3)}^1{\times x}^{7-1}+\left(\begin{array}{c}7\\ 2\end{array}\right)\times {(-3)}^2\times x^{7-2}+.....+\left(\begin{array}{c}7\\ 6\end{array}\right)\times 3^6{\times \left(2x\right)}^{9-8}+\left(\begin{array}{c}7\\ 7\end{array}\right)\times {(-3)}^7 \\ =\left(\begin{array}{c}7\\ 0\end{array}\right)x^7-\left(\begin{array}{c}7\\ 1\end{array}\right)3x^6+\left(\begin{array}{c}9\\ 2\end{array}\right)9x^5+....+\left(\begin{array}{c}7\\ 6\end{array}\right)729x-\left(\begin{array}{c}7\\ 7\end{array}\right)2187 \end{gathered}$$

Step 3. Description of finding the third term

The third term from the expanded form is$$\begin{gathered} \left(\begin{array}{c}9\\ 2\end{array}\right)9x^5=\frac{9!}{2!7!}\times 9\times x^5 \\ =\frac{9\times 8\times 7!}{2\times 7!}\times 9\times x^5 \\ =324x^5 \end{gathered}$$