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:

For binomial expansions, focus on the terms that result in the required power and ignore others. Multiply the terms to get the desired coefficient.

Answer 1

The Correct Option is A

- Expand \((x - 1)^3\) and \((x - 2)^3\) using the binomial expansion.
\[ (x - 1)^3 = x^3 - 3x^2 + 3x - 1 \] \[ (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \] - Now multiply the two expanded polynomials: \[ (x^3 - 3x^2 + 3x - 1)(x^3 - 6x^2 + 12x - 8). \] - To find the coefficient of \(x^4\), focus on the terms that contribute to \(x^4\). These are:
\(x^3 \times -6x^2 = -6x^5\),
\(-3x^2 \times 12x = -36x^3\),
\(3x \times -6x^2 = -18x^3\),
\(-1 \times x^3 = -x^3\).
- The coefficient of \(x^4\) is the result of the product of the relevant terms, which gives: \[ \boxed{33}. \]