GATE 2023 PH – Question 9
The coefficient of $x^4$ in the polynomial $(x-1)^3(x-2)^3$ is equal to _______.
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Correct answer: (A) 33
Explanation
**Step 1: expand each factor.**
$$(x-1)^3=x^3-3x^2+3x-1,\qquad(x-2)^3=x^3-6x^2+12x-8 .$$
**Step 2: collect the $x^4$ terms** from the product of the two cubics. The pairs of powers adding to 4 are $(3,1)$, $(2,2)$ and $(1,3)$:
- $x^3\cdot12x=12x^4$
- $(-3x^2)(-6x^2)=18x^4$
- $(3x)\cdot x^3=3x^4$
**Step 3: add.**
$$12+18+3=\mathbf{33}.$$