Discussion
If \(f(x) = ax + bx^{2}\) then the co-efficient of \(x^{3}\) in \(f(f(x))\) is
Answer: E
The coefficient of \(x^{3}\) is \(2ab^{2}\).
Substitute \(f(x)\) into itself:
\[ \begin{aligned} f(f(x)) &= a(ax + bx^{2}) + b(ax + bx^{2})^{2} \\ &= a^{2}x + abx^{2} + b(a^{2}x^{2} + 2abx^{3} + b^{2}x^{4}) \end{aligned} \]
The only \(x^{3}\) term comes from the middle term of the square, \(b \times 2abx^{3} = 2ab^{2}x^{3}\). \(ab^{2}\) forgets the factor 2 in \((p + q)^{2} = p^{2} + 2pq + q^{2}\); \(ab\) is the coefficient of one of the \(x^{2}\) terms, and \(a^{3}\) does not occur.
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