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 avoid errors, list the powers of \(x\) in the first polynomial (e.g., 3, 2, 1, 0) and find the corresponding power needed from the second polynomial to sum to the target power (e.g., to get 4, you need 1, 2, 3, 4 respectively). Then multiply the coefficients for each valid pair.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the coefficient of a specific term (like \(x^4\)) in the product of two polynomials, we do not need to expand the entire product. We only need to find the pairs of terms, one from each polynomial, whose product results in the desired power of \(x\), and then sum their coefficients.
Step 2: Key Formula or Approach:
We will use the binomial expansion formula: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).
First, we expand both \((x - 1)^3\) and \((x - 2)^3\).
Then, we identify the combinations of terms that multiply to give an \(x^4\) term.
Step 3: Detailed Calculation:
Expansion of the polynomials:
1. \((x - 1)^3 = x^3 - 3(x^2)(1) + 3(x)(1^2) - 1^3 = \mathbf{x^3 - 3x^2 + 3x - 1}\) 2. \((x - 2)^3 = x^3 - 3(x^2)(2) + 3(x)(2^2) - 2^3 = \mathbf{x^3 - 6x^2 + 12x - 8}\) Finding the \(x^4\) terms:
We need to multiply the two expanded polynomials: \((x^3 - 3x^2 + 3x - 1)(x^3 - 6x^2 + 12x - 8)\). We find pairs of terms (one from each polynomial) whose powers of \(x\) add up to 4.
- (Term with \(x^3\) from the first) \(\times\) (Term with \(x^1\) from the second):
\[ (x^3) \times (12x) = 12x^4 \] - (Term with \(x^2\) from the first) \(\times\) (Term with \(x^2\) from the second):
\[ (-3x^2) \times (-6x^2) = 18x^4 \] - (Term with \(x^1\) from the first) \(\times\) (Term with \(x^3\) from the second):
\[ (3x) \times (x^3) = 3x^4 \] - (Term with \(x^0\) from the first) \(\times\) (Term with \(x^4\) from the second):
There is no \(x^4\) term in the second polynomial, so this combination is not possible.
Summing the coefficients:
The total coefficient of \(x^4\) is the sum of the coefficients from the products we found. \[ \text{Coefficient} = 12 + 18 + 3 = 33 \] Step 4: Final Answer:
The coefficient of \(x^4\) is 33.
Step 5: Why This is Correct:
The solution correctly expands the cubic terms and systematically identifies all pairs of terms whose product yields \(x^4\). The sum of the coefficients of these products gives the final coefficient. The calculation is accurate and leads to the correct option (A).