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

Hint:

To get a specific coefficient in a product, convolve the coefficients: only pairs whose degrees add to the target degree contribute.

Answer 1

The Correct Option is A

Expand each cubic: $(x-1)^3=x^3-3x^2+3x-1$, $(x-2)^3=x^3-6x^2+12x-8$.
Let $A(x)=\sum_{i=0}^{3}a_i x^i$ with $(a_0,a_1,a_2,a_3)=(-1,3,-3,1)$, and $B(x)=\sum_{j=0}^{3}b_j x^j$ with $(b_0,b_1,b_2,b_3)=(-8,12,-6,1)$.
In the product $A(x)B(x)$, the coefficient of $x^4$ is $\sum_{i+j=4} a_i b_j = a_1b_3+a_2b_2+a_3b_1$.
Compute: $a_1b_3=3. 1=3$, $a_2b_2=(-3).(-6)=18$, $a_3b_1=1. 12=12$.
Sum: $3+18+12=33$.
\[ \boxed{33} \]