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 $n$.