Multiplicity

Definition

Multiplicity of a root in a polynomial function is the number of times that root occurs. It affects the shape and behavior of the graph at the corresponding $x$-intercept.

5 Must Know Facts For Your Next Test

1. The multiplicity of a root determines whether the graph touches or crosses the x-axis at that root.
2. A root with an odd multiplicity will cause the graph to cross the x-axis.
3. A root with an even multiplicity will cause the graph to touch but not cross the x-axis.
4. The sum of all multiplicities in a polynomial equals its degree.
5. Multiplicity influences local maxima, minima, and points of inflection near roots.

Review Questions

• What is the effect on a polynomial's graph if a root has an odd multiplicity?
• How does the graph behave at a root with even multiplicity?
• If a polynomial has roots 2 (multiplicity 3) and -1 (multiplicity 1), what is its degree?

Related terms

Root: A solution to the equation $P(x) = 0$, where $P(x)$ is a polynomial function.

Degree: The highest power of $x$ in a polynomial function.

X-intercept: The point(s) where a graph intersects the x-axis, corresponding to real roots of the function.