GATE CH 2023 | GA Question: 9

Question

The coefficient of $x^{4}$ in the polynomial $(x-1)^{3}(x-2)^{3}$ is equal to___________.

• A. $33$
• B. $-3$
• C. $30$
• D. $21$

Answer 1

We know that, $(a-b)^3 = a^3 – b^3 -3ab(a-b)$

Now,

$(x-1)^3 = x^3-1^3-3x(x-1) =  x^3-1-3x^2+3x = x^3-3x^2 + 3x -1$

$(x-2)^3 = x^3-2^3-6x(x-2) =  x^3-8-6x^2 + 12x = x^3-6x^2+12x-8$

Multiply both the terms and just focus on to make $x^4.$

$(x-1)^3 (x-2)^3 = (x^3-3x^2 + 3x -1)( x^3-6x^2+12x-8)$

$\qquad \qquad  = 12x^4 + 18x^4 + 3x^4 = 33x^4$

$\therefore$ The coefficient of $x^4 = 33.$

Correct Answer: A