The expression \((x - 3)^2 \cdot (x + 1)^3\) is already in factored form. If you want to expand it, you can use the binomial theorem for each part.

Let’s expand this step-by-step:

### Step 1: Expand \((x - 3)^2\)
\[
(x - 3)^2 = x^2 - 6x + 9
\]

### Step 2: Expand \((x + 1)^3\)
\[
(x + 1)^3 = x^3 + 3x^2 + 3x + 1
\]

### Step 3: Multiply the two expanded forms
Now we have:
\[
(x - 3)^2 \cdot (x + 1)^3 = (x^2 - 6x + 9)(x^3 + 3x^2 + 3x + 1)
\]

Expanding this product term-by-term:
1. \(x^2 \cdot x^3 = x^5\)
2. \(x^2 \cdot 3x^2 = 3x^4\)
3. \(x^2 \cdot 3x = 3x^3\)
4. \(x^2 \cdot 1 = x^2\)

5. \(-6x \cdot x^3 = -6x^4\)
6. \(-6x \cdot 3x^2 = -18x^3\)
7. \(-6x \cdot 3x = -18x^2\)
8. \(-6x \cdot 1 = -6x\)

9. \(9 \cdot x^3 = 9x^3\)
10. \(9 \cdot 3x^2 = 27x^2\)
11. \(9 \cdot 3x = 27x\)
12. \(9 \cdot 1 = 9\)

### Step 4: Combine all terms
Now add together all terms by degree:
\[
x^5 + (3x^4 - 6x^4) + (3x^3 - 18x^3 + 9x^3) + (x^2 - 18x^2 + 27x^2) + (-6x + 27x) + 9
\]

Simplifying:
\[
= x^5 - 3x^4 - 6x^3 + 10x^2 + 21x + 9
\]

So, the expanded form is:
\[
(x - 3)^2 \cdot (x + 1)^3 = x^5 - 3x^4 - 6x^3 + 10x^2 + 21x + 9
\]

Would you like more details on any step?

---

### Related Questions:
1. How do we expand binomials raised to a power using the binomial theorem?
2. How do we multiply polynomials systematically to avoid missing terms?
3. What are common mistakes to avoid when expanding products of polynomials?
4. How can we verify the expansion result by factoring it back?
5. How does expanding polynomials help in calculus, especially for differentiation?

### Tip:
When expanding products of polynomials, it's helpful to align terms by their degree to avoid errors and simplify addition afterward.

Explore the step-by-step solution for expanding the polynomial expression (x-3)^2 * (x+1)^3 using the binomial theorem and polynomial multiplication. This guide provides detailed steps, including individual expansions and combining terms for simplification. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation