# Thread: State the multiplicity of each zero

1. ## State the multiplicity of each zero

Find the zeros of the polynomial function and state the multiplicity of each zero:

$\displaystyle P(x)=x^2 (3 x+5)^2$

So I know that $\displaystyle x=-5/3$ but sense things are squared I'm loosing the picture.

Is it 0 (multiplicity of 2) and $\displaystyle -5/3$ (multiplicity of 2)?

2. Originally Posted by RBlax
Is it 0 (multiplicity of 2) and $\displaystyle -5/3$ (multiplicity of 2)?
Yup.

3. I honestly can just see it, but don't know the math that makes it true.

How do I prove this?

4. Originally Posted by RBlax
I honestly can just see it, but don't know the math that makes it true.

How do I prove this?
You don't need to do any computation for this problem (other than solving 3x + 5 = 0), you just need to be able to apply the definition of multiplicity.

Your book probably has a definition. Here is one definition:

For a polynomial $\displaystyle P(x)$ with root $\displaystyle a$, the multiplicity of $\displaystyle a$ is the maximum integer $\displaystyle k$ for which $\displaystyle (x - a)^k$ divides $\displaystyle P(x)$.

If you wanted to, you could re-express $\displaystyle P(x)$ given above as

$\displaystyle P(x)=9x^2\left(x-\left(-\frac{5}{3}\right)\right)^2$