Discussion
If the coefficient of \(x^{4}\) in the binomial expansion of \((4x + a)^{7}\) is -1120 , then the value of a is equal to
Answer: C
a = \(-\frac{1}{2}\).
The general term of \((4x + a)^{7}\) is \(\binom{7}{r}(4x)^{7 - r}a^{r}\). The \(x^{4}\) term has 7 − r = 4, so r = 3.
\[ \begin{aligned} \binom{7}{3}4^{4}a^{3} &= -1120 \\ 35 \times 256\,a^{3} &= -1120 \\ a^{3} &= -\frac{1}{8} \\ a &= -\frac{1}{2} \end{aligned} \]
The negative coefficient needs a negative a, which rules out the positive options; \(-\frac{1}{8}\) is \(a^{3}\), not a.
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