Multiplicity

Definition

Multiplicity refers to the number of times a particular root is repeated in a polynomial function. It affects the shape and behavior of the graph at that root.

5 Must Know Facts For Your Next Test

1. A root's multiplicity determines whether the graph touches or crosses the x-axis at that point.
2. If a root has even multiplicity, the graph will touch but not cross the x-axis at that point.
3. If a root has odd multiplicity, the graph will cross the x-axis at that point.
4. The sum of the multiplicities of all roots equals the degree of the polynomial.
5. Multiplicity impacts how 'flat' or 'steep' the graph appears near its roots.

Review Questions

• How does the multiplicity of a root affect whether a polynomial's graph touches or crosses the x-axis?
• What can be inferred about a polynomial if one of its roots has an even multiplicity?
• Explain how you would determine if a polynomial's degree is consistent with its given roots and their multiplicities.

Related terms

Root: A value for which a given polynomial equals zero.

Degree: The highest power of the variable in a polynomial function.

Factorization: \text{The process of breaking down a polynomial into products of simpler polynomials.}