# Zeros and Multiplicities of Factored Polynomials

Zeros and Multiplicities of Factored Polynomials

This Demonstration explores the zeros of polynomials and their multiplicities on a graph. The zeros of a function are those values of for which . On the graph, the zeros give the locations of the intercepts.

y=f(x)

x

y=0

x

The factor theorem states that is a zero of a polynomial if and only if is a factor of that polynomial, i.e. if and only if for some other polynomial . With that in mind, the multiplicity of a zero denotes the number of times that appears as a factor. For example, has a zero at of multiplicity 6. On the graph, the multiplicity of a zero tells you whether the function crosses the axis (in the case that the multiplicity is odd) or simply touches the axis (in the case that the multiplicity is even).

r

p(x)

x-r

p(x)=q(x)(x-r)

q(x)

r

x-r

y=

6

(x-2)

x=2