Find the roots of the polynomial function state the multiplicity of each roots 

To find the roots of a polynomial function and determine their multiplicities, follow these steps:

1. **Factor the polynomial completely.**
2. **Set each factor equal to zero** and solve for the variable to find the roots.
3. **Determine the multiplicity** of each root by identifying the exponent of the corresponding factor.

Let's work through an example. Consider the polynomial:

\[ f(x) = (x - 2)^2(x + 3) \]

### Step 1: Identify the factors
The polynomial is already factored as \( (x - 2)^2(x + 3) \).

### Step 2: Set each factor equal to zero
- \( (x - 2)^2 = 0 \) gives the root \( x = 2 \).
- \( (x + 3) = 0 \) gives the root \( x = -3 \).

### Step 3: Determine the multiplicity
- The factor \( (x - 2)^2 \) indicates that the root \( x = 2 \) has **multiplicity 2** (since the exponent is 2).
- The factor \( (x + 3) \) has no exponent, so the root \( x = -3 \) has **multiplicity 1**.

### Final answer:
- Root \( x = 2 \) has multiplicity 2.
- Root \( x = -3 \) has multiplicity 1.

Would you like me to work through a different example, or clarify any part of this process?

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Here are 5 related questions to expand your understanding:
1. How do multiplicities affect the shape of a polynomial graph?
2. What happens to the graph at roots with odd multiplicities?
3. Can a polynomial function have complex roots? How do they appear?
4. What is the relationship between the degree of the polynomial and the number of roots?
5. How can you check if you've found all the roots of a polynomial?

**Tip**: For a polynomial of degree \(n\), the sum of the multiplicities of all roots must equal \(n\).

Find the roots of the polynomial function and state the multiplicity of each root.

Learn how to find the roots of a polynomial function and determine their multiplicities. This example covers factoring polynomials and applying the Fundamental Theorem of Algebra. Ideal for students in Grades 10-12.

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