explain how to identify the degree of a function with only the roots?

To identify the degree of a polynomial function using only the roots, follow these steps:

1. **Count the Number of Distinct Roots**: The degree of the polynomial is at least as large as the number of distinct roots. For example, if a polynomial has 3 distinct roots, the degree is at least 3.

2. **Consider Multiplicity of the Roots**: If any of the roots are repeated, you must take into account the **multiplicity** of those roots. The multiplicity of a root refers to how many times that root is repeated. For example, if a polynomial has a root at \( x = 2 \) with multiplicity 3, this means the root \( x = 2 \) appears three times, contributing three to the degree of the polynomial.

3. **Add Up the Multiplicities**: The degree of the polynomial is the sum of the multiplicities of all the roots. For instance, if you have the roots \( x = 1 \) with multiplicity 2, and \( x = -3 \) with multiplicity 1, then the degree of the polynomial is \( 2 + 1 = 3 \).

### Example:
Consider a polynomial with roots \( x = 2 \) (with multiplicity 2), \( x = -1 \) (with multiplicity 1), and \( x = 4 \) (with multiplicity 1).
- The total degree of the polynomial is \( 2 + 1 + 1 = 4 \).

Thus, this polynomial is of degree 4.

Would you like any further details on how this works or an example problem?

### Related Questions:
1. How can we find the polynomial from its roots and degree?
2. What role does the leading coefficient play in determining the degree of a polynomial?
3. How can you determine the multiplicity of a root if the function isn't factored?
4. What is the relationship between the number of turning points and the degree of a polynomial?
5. Can non-polynomial functions have roots, and how are those related to their structure?

**Tip**: The degree of a polynomial tells you the maximum number of solutions (or roots) it can have, considering multiplicities.

Discover how to identify the degree of a polynomial function based on its roots, including distinct roots and their multiplicities. Learn key algebraic concepts and the fundamental theorem of algebra. Suitable for high school students in Grades 9-12.

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