Polynomial $x^4$ Coefficient Calculation

We need to determine the coefficient of the $x^4$ term in the expansion of the polynomial $(x - 1)^3(x – 2)^3$.

Step 1: Expand $(x - 1)^3$

Using the binomial expansion $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$: $(x - 1)^3 = x^3 - 3(x^2)(1) + 3(x)(1)^2 - 1^3$ $(x - 1)^3 = x^3 - 3x^2 + 3x - 1$

Step 2: Expand $(x - 2)^3$

Using the same binomial expansion formula: $(x - 2)^3 = x^3 - 3(x^2)(2) + 3(x)(2)^2 - 2^3$ $(x - 2)^3 = x^3 - 6x^2 + 12x - 8$

Step 3: Identify $x^4$ terms in the product

Let $P(x) = x^3 - 3x^2 + 3x - 1$ and $Q(x) = x^3 - 6x^2 + 12x - 8$. We multiply $P(x)$ by $Q(x)$. To find the $x^4$ term, we identify pairs of terms, one from each polynomial, whose powers sum to 4:

• From $P(x)$'s $x^3$ term and $Q(x)$'s $x^1$ term: $(x^3) \times (12x) = 12x^4$. The coefficient is $1 \times 12 = 12$.
• From $P(x)$'s $x^2$ term and $Q(x)$'s $x^2$ term: $(-3x^2) \times (-6x^2) = 18x^4$. The coefficient is $(-3) \times (-6) = 18$.
• From $P(x)$'s $x^1$ term and $Q(x)$'s $x^3$ term: $(3x) \times (x^3) = 3x^4$. The coefficient is $3 \times 1 = 3$.

Step 4: Sum the coefficients

The total coefficient for the $x^4$ term is the sum of the coefficients found in Step 3: $12 + 18 + 3 = 33$

Final Answer

The coefficient of $x^4$ in the polynomial $(x - 1)^3(x – 2)^3$ is 33.