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:

- Use the binomial expansion to expand polynomials and identify the terms that contribute to specific powers of \( x \).

Answer 1

The Correct Option is A

We need to find the coefficient of \( x^4 \) in the product \( (x - 1)^3 (x - 2)^3 \). First, expand both binomials using the binomial theorem:
\[ (x - 1)^3 = x^3 - 3x^2 + 3x - 1 \] \[ (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \] Now, multiply the two expanded expressions:
\[ (x - 1)^3 (x - 2)^3 = (x^3 - 3x^2 + 3x - 1)(x^3 - 6x^2 + 12x - 8) \] The term that produces \( x^4 \) comes from multiplying the \( x^3 \) term from the first binomial by the \( -6x^2 \) term from the second binomial:
\[ x^3 \times (-6x^2) = -6x^5 \] Now, multiply the remaining terms and combine like terms. The coefficient of \( x^4 \) is 33. Final Answer: The coefficient of \( x^4 \) is \( \boxed{33} \).