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:

When finding coefficients in product expansions, focus on degree combinations that add to the required power.
Avoid expanding everything; just target terms that contribute to the desired power.

Answer 1

The Correct Option is A

Step 1: Expand \((x-1)^3 = x^3 - 3x^2 + 3x - 1\).
Step 2: Expand \((x-2)^3 = x^3 - 6x^2 + 12x - 8\).
Step 3: To find the coefficient of \(x^4\), consider cross terms from the two cubic polynomials whose degrees add to 4.
\[ \begin{aligned} & (x^3 \cdot -6x^2) \quad \Rightarrow -6x^5 \quad (\text{too high})
& (x^3 \cdot 12x) = 12x^4
& (x^3 \cdot -8) = -8x^3 \quad (\text{too low})
& (-3x^2 \cdot x^3) = -3x^5 \quad (\text{too high})
& (-3x^2 \cdot -6x^2) = 18x^4
& (3x \cdot x^3) = 3x^4
& (-1 \cdot x^3) = -x^3 \quad (\text{too low}) \end{aligned} \] Step 4: Adding valid contributions for \(x^4\): \(12 + 18 + 3 = 33\). Wait—check carefully.
Actually, recheck carefully:
From \((x-1)^3(x-2)^3\): expand term by term:
Coefficient of \(x^4\) arises from: \[ \begin{aligned} & (x^3 \cdot 12x) = 12x^4
& (-3x^2 \cdot -6x^2) = 18x^4
& (3x \cdot x^3) = 3x^4 \end{aligned} \] Total = \(12+18+3=33\).
Step 5: So coefficient of \(x^4\) is \(\boxed{33}\). Hence answer is (A).