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) $1,172$

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 $\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^{n-k}{b}^k$ , 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 $\left(\genfrac{}{}{0ex}{}{8}{4}\right)\cdot (x^2{)}^{8-4}\cdot (-2{)}^4=\left(\genfrac{}{}{0ex}{}{8}{4}\right)\cdot 16$ . Calculating $\left(\genfrac{}{}{0ex}{}{8}{4}\right)=\frac{8!}{4!4!}=70$ , the coefficient is $70\cdot 16=1120$ .