Question

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

The coefficient of x⁷ in the expansion of (2x - 1)¹²

Short answer

The coefficient of$x^{7}in{(2x-1)}^{12}is-101376$

Step-by-step solution

. Given information

${(2x-7)}^{12}$is given.

We have to find the coefficient of term x⁷

 step 2 . Expansion of  (2x-1)12  using The Binomial Theorem

The binomial theorem state 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 (2x-12)¹² here a= -1 and n=12 ,then, we get $$\begin{gathered} {(2x-1)}^{12}=\left(\begin{array}{c}12\\ 0\end{array}\right){\left(2x\right)}^{12}+\left(\begin{array}{c}12\\ 1\end{array}\right)(-1){\left(2x\right)}^{12-1}+\left(\begin{array}{c}12\\ 2\end{array}\right){(-1)}^2{\left(2x\right)}^{12-2}+\left(\begin{array}{c}12\\ 3\end{array}\right){(-1)}^3{\left(2x\right)}^{12-3}+\left(\begin{array}{c}12\\ 4\end{array}\right){(-1)}^4{\left(2x\right)}^{12-4}+\left(\begin{array}{c}12\\ 5\end{array}\right){(-1)}^5{\left(2x\right)}^{12-5}+......+\left(\begin{array}{c}12\\ 11\end{array}\right){(-1)}^{11}{\left(2x\right)}^{12-11}+\left(\begin{array}{c}12\\ 12\end{array}\right)1^{12} \\ =\left(\begin{array}{c}12\\ 0\end{array}\right){\left(2x\right)}^{12}-\left(\begin{array}{c}12\\ 1\end{array}\right){\left(2x\right)}^{11}+\left(\begin{array}{c}12\\ 2\end{array}\right){\left(2x\right)}^{10}-\left(\begin{array}{c}12\\ 3\end{array}\right){\left(2x\right)}^9+\left(\begin{array}{c}12\\ 4\end{array}\right){\left(2x\right)}^8-\left(\begin{array}{c}12\\ 5\end{array}\right){\left(2x\right)}^7+......-\left(\begin{array}{c}12\\ 11\end{array}\right){\left(2x\right)}^1+\left(\begin{array}{c}12\\ 12\end{array}\right) \end{gathered}$$

Step 3. Description of finding the coefficient of x7

According to the question, the coefficient of x⁷ from the expanded term is

$$\begin{gathered} \left(\begin{array}{c}12\\ 5\end{array}\right){\left(2\right)}^7{(-1)}^5=\frac{12!}{5!7!}\times 2^7\times -1 \\ =-\frac{12\times 11\times 10\times 9\times 8\times 7!}{5\times 4\times 3\times 2\times 1\times 7!}\times 128 \\ =-101376,thisisthecoe.of{x}^7 \end{gathered}$$