Question

The coefficient of \(x^4\) in \((x-1)^3(x-2)^3\) is equal to \underline{\hspace{1cm}}.

• A. 33
• B. \(-3\)
• C. 30
• D. 21

Hint:

To find a single coefficient in a product, avoid full expansion. Identify term pairs whose degrees add to the target degree and sum their coefficient products (a discrete convolution).

Answer 1

The Correct Option is A

Step 1: Expand each cubic (or recall coefficients).
\((x-1)^3 = x^3 - 3x^2 + 3x - 1\).
\((x-2)^3 = x^3 - 6x^2 + 12x - 8\).

Step 2: Collect contributions to the \(x^4\) term.
In the product, \(x^4\) arises from pairs whose degrees sum to \(4\):
\(\bullet\) \(x^3\) (from first) \(\times\) \(x\) (from second): \(1 \cdot 12 = 12\).
\(\bullet\) \(x^2\) (from first, coeff \(-3\)) \(\times\) \(x^2\) (from second, coeff \(-6\)): \((-3)(-6)=18\).
\(\bullet\) \(x\) (from first, coeff \(3\)) \(\times\) \(x^3\) (from second, coeff \(1\)): \(3\cdot1=3\).
No \(x^4\) term exists in either factor, so no other contributions.

Step 3: Sum the contributions.
\[ 12 + 18 + 3 = 33 \;\Rightarrow\; \text{coefficient of } x^4 = 33. \]

Final Answer:
\[ \boxed{33} \]