Discussion
Let \((1 + ax)(1 - 2x)^{3} = \sum_{n=0}^{4} a_{n}x^{n}\) , where \(a\) is a constant. If \(a_{2} = 0\) , then the value of \(a\) is
Answer: E
The value of a is 2.
Expand \((1 - 2x)^{3} = 1 - 6x + 12x^{2} - 8x^{3}\). In the product with \((1 + ax)\), the \(x^{2}\) terms come from \(1 \times 12x^{2}\) and \(ax \times (-6x)\).
\[ \begin{aligned} a_2 &= 12 - 6a = 0 \\ a &= 2 \end{aligned} \]
Taking the \(x^{2}\) coefficient as 12 alone, or pairing ax with the \(x^{3}\) term, ignores how the product forms \(x^{2}\); only \(12 - 6a = 0\) uses both contributions.
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