Find the coefficient of the term containing $x ^ { 8 }$ in the expansion of the binomial $\left( x ^ { 2 } - 2 \right) ^ { 8 }$ .
A) $1,188$
B) $1,132$
C) $1,200$
D) $1,120$
E) $1,172$

Question

Quizplus · Accepted Answer

The coefficient of $x^8$ in the expansion of $\left(x^2 - 2\right)^8$ can be found using the binomial theorem, specifically the term where $x^2$ is raised to the 4th power (since $2 \times 4 = 8$). The general term in a binomial expansion is given by $\binom{n}{k}a^{n-k}b^k$, where $n$ is the power of the binomial, $a$ and $b$ are the terms of the binomial, and $k$ is the term number. For the term containing $x^8$, $k = 4$, so the coefficient is $\binom{8}{4} \cdot (x^2)^{8-4} \cdot (-2)^4 = \binom{8}{4} \cdot 16$. Calculating $\binom{8}{4} = \frac{8!}{4!4!} = 70$, the coefficient is $70 \cdot 16 = 1120$.

Find the Coefficient of the Term Containing x8x ^ { 8 }x8

Find the Coefficient of the Term Containing $x ^ { 8 }$