The GATE Grind

GATE 2023 CH – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

The coefficient of $x^4$ in the polynomial $(x - 1)^3(x - 2)^3$ is equal to _______.

  1. 33
  2. −3
  3. 30
  4. 21

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) 33

Explanation

The product is $\left[(x - 1)(x - 2)\right]^3 = (x^2 - 3x + 2)^3$. The $x^4$ terms arise from taking $x^2, x^2, 2$ in 3 ways, giving $3 \times 2 = 6$, and from $x^2, -3x, -3x$ in 3 ways, giving $3 \times 9 = 27$. The total is $6 + 27 = 33$.