Discussion
The coefficient of \(x^{4}\) in the polynomial \((x - 1)^{3}(x - 2)^{3}\) is equal to _______.
Answer: A
The coefficient is 33.
Write the product as a cube: \((x - 1)(x - 2) = x^{2} - 3x + 2\), so the polynomial is \((x^{2} - 3x + 2)^{3}\). An \(x^{4}\) term comes from choosing one term from each of the three factors in two ways:
\[ \begin{aligned} x^{2} \cdot x^{2} \cdot 2 &: \ 3 \times 2 = 6 \\ x^{2} \cdot (-3x) \cdot (-3x) &: \ 3 \times 9 = 27 \\ \text{total} &= 6 + 27 = 33 \end{aligned} \]
Each case has 3 arrangements, because the odd factor out can be any of the three. Taking \((-3)^{2}\) as -9 gives 6 - 27 = -21, whose size is the distractor 21, and counting each case only once gives 2 + 9 = 11, so the factor 3 for the arrangements matters.
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