Advertisement. Polynomials have the property that if r is a zero of the polynomial p (x), then p (x) is divisible by x−r. The multiplicity of a zero is how many times you can divide out

Solve Now
The multiplicity of a zero z of a function f is the number n such that lim x → z f ( x) ( x − z) n is finite, providing that the limit exists. (By finite, I mean not zero and not infinite.) Of course it is

The multiplicity of each zero is the number of times that its corresponding factor appears. In other words, the multiplicities are the powers. (For the factor x − 5 , the understood power is 1 .)