Question Statement:
The coefficient of x4 in the polynomial (x - 1)3(x - 2)3 is equal to _______.
Options:
-
(A) 33
-
(B) ̶ 3
-
(C) 30
-
(D) 21
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2023 Q.9)**
**1. Governing Concept**:
- **Topic**: Polynomial Expansions & Binomial Theorem
- Given expression:
$P(x) = (x - 1)^3 (x - 2)^3 = [(x - 1)(x - 2)]^3 = [x^2 - 3x + 2]^3$
- Expanding $(x^2 - 3x + 2)^3$ via multinomial expansion:
$(a + b + c)^3 = \sum \frac{3!}{i! j! k!} a^i b^j c^k \quad \text{with } i + j + k = 3$
where $a = x^2, b = -3x, c = 2$.
- Power of $x$ is $2i + j = 4$:
- Case 1: $i = 2, j = 0, k = 1 \implies \frac{6}{2!0!1!} (x^2)^2 (-3x)^0 (2)^1 = 3 \cdot 1 \cdot 1 \cdot 2 \cdot x^4 = 6x^4$
- Case 2: $i = 1, j = 2, k = 0 \implies \frac{6}{1!2!0!} (x^2)^1 (-3x)^2 (2)^0 = 3 \cdot x^2 \cdot 9x^2 \cdot 1 = 27x^4$
- Total coefficient of $x^4$:
$6 + 27 = 33$
**2. Option Analysis**:
- **Option (A) [33] is CORRECT**: Verified by rigorous multinomial expansion ($6 + 27 = 33$).
- **Option (B) is INCORRECT**: Negative term.
- **Option (C) [30] is INCORRECT**: Missing minor product term.
- **Option (D) [21] is INCORRECT**: Calculation mistake.
**3. Final Answer**:
- **Correct Choice**: **A**
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