Recall that multiplicity is how many times the root or factor appears. What is the multiplicity of each root? How do you know?

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To determine the multiplicity of each root of a polynomial, we need to follow these steps:

1. **Identify the roots of the polynomial**: This can be done by factoring the polynomial or using the quadratic formula for quadratic equations.

2. **Determine the factors**: Each root corresponds to a factor of the polynomial. The multiplicity of a root is the exponent of its corresponding factor when the polynomial is expressed in its factored form.

3. **Count the occurrences**: The multiplicity is the number of times a particular root appears as a factor.

### Example

Let's consider a polynomial $$ P(x) = (x - 2)^3 (x + 1)^2 (x - 3) $$.

1. **Identify the roots**:
   - The roots are $$ x = 2 $$, $$ x = -1 $$, and $$ x = 3 $$.

2. **Determine the factors**:
   - The factor $$ (x - 2) $$ appears with an exponent of 3.
   - The factor $$ (x + 1) $$ appears with an exponent of 2.
   - The factor $$ (x - 3) $$ appears with an exponent of 1.

3. **Count the occurrences**:
   - The root $$ x = 2 $$ has a multiplicity of 3.
   - The root $$ x = -1 $$ has a multiplicity of 2.
   - The root $$ x = 3 $$ has a multiplicity of 1.

### Conclusion

The multiplicity of each root is:
- $$ x = 2 $$: multiplicity 3
- $$ x = -1 $$: multiplicity 2
- $$ x = 3 $$: multiplicity 1

You can determine the multiplicity by examining the factored form of the polynomial and counting the exponents of each factor corresponding to the roots.
Each root's multiplicity is determined by the exponent of its corresponding factor in the polynomial's factored form. For example, in $$ P(x) = (x - 2)^3 (x + 1)^2 (x - 3) $$, the root $$ x = 2 $$ has a multiplicity of 3, $$ x = -1 $$ has a multiplicity of 2, and $$ x = 3 $$ has a multiplicity of 1.

Recall that multiplicity is how many times the root or factor appears. What is the multiplicity of each root? How do you know?

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