Choose the one alternative that best completes the statement or answers the question.
-The coefficient of $x ^ { 8 }$ in the expansion of $\left( x ^ { 2 } - 3 \right) ^ { 7 }$
A) 945
B) $- 945$
C) 2835
D) $- 2835$

Question

Quizplus · Accepted Answer

The coefficient of $x^8$ in the expansion of $\left(x^2 - 3\right)^7$ can be found using the binomial theorem. The general term in the expansion of $(a + b)^n$ is given by $\binom{n}{r}a^{n-r}b^r$, where $r$ is the term number starting from 0. To find the coefficient of $x^8$, we need the term with $x^{8}$, which comes from $x^2$ raised to the 4th power (since $2 \times 4 = 8$), and $-3$ raised to the 3rd power. Thus, we calculate the coefficient as $\binom{7}{3}(x^2)^4(-3)^3$. The binomial coefficient $\binom{7}{3} = 35$, and $(-3)^3 = -27$, so the coefficient of $x^8$ is $35 \times 1 \times -27 = -945$.

Choose the One Alternative That Best Completes the Statement or Answers