Question

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

The coefficient of x⁷ in the expansion of (2x +3)⁹

Short answer

The coefficient of x⁷ is 4608

Step-by-step solution

. Given information

Here (2x+3)⁹ is given.

we have to find out the coefficient of the term x⁷

. Expansion of  (2x+3)9 using The Binomial Theorem

According to the binomial theorem

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

Based on this we can expand, here a=1 and n=9

localid="1647339902877" $$\begin{gathered} {(2x+3)}^9=\left(\begin{array}{c}9\\ 0\end{array}\right){\left(2x\right)}^9+\left(\begin{array}{c}9\\ 1\end{array}\right)\left(1\right){\left(2x\right)}^{9-1}+\left(\begin{array}{c}9\\ 2\end{array}\right){\left(1\right)}^2{\left(2x\right)}^{9-2}+\left(\begin{array}{c}9\\ 3\end{array}\right){\left(1\right)}^3{\left(2x\right)}^{9-3}+......+\left(\begin{array}{c}9\\ 8\end{array}\right){\left(1\right)}^8{\left(2x\right)}^{9-8}+\left(\begin{array}{c}9\\ 9\end{array}\right)1^9 \\ =\left(\begin{array}{c}9\\ 0\end{array}\right){\left(2x\right)}^9+\left(\begin{array}{c}9\\ 1\end{array}\right){\left(2x\right)}^8+\left(\begin{array}{c}9\\ 2\end{array}\right){\left(2x\right)}^7+\left(\begin{array}{c}9\\ 3\end{array}\right){\left(2x\right)}^6+......+\left(\begin{array}{c}9\\ 8\end{array}\right){\left(2x\right)}^1+\left(\begin{array}{c}9\\ 9\end{array}\right) \end{gathered}$$

Step 3. Description of finding the coefficient of x7

From the expanded form of (2x+3)⁹ we get a coefficient of x⁷

then the coefficient of x⁷ is

$$\begin{gathered} \left(\begin{array}{c}9\\ 2\end{array}\right){\left(2\right)}^7{\left(1\right)}^2,whichisequalto\frac{9!}{2!7!}\times 2^7=\frac{9\times 8\times 7!}{2\times 1\times 7!}\times 128 \\ =4608 \end{gathered}$$