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.