# How do you write a polynomial with zeros -2,-2,3,-4i?

Mar 8, 2016

$f \left(x\right) = {\left(x + 2\right)}^{2} \left(x - 3\right) \left(x + 4 i\right)$

#### Explanation:

Well, we know that a polynomial with four roots (or four zeros, if you will) is a polynomial of degree four. That means that that polynomial can be expressed as the product of four monomials.

We also know that if we hit any of the zeros, the entire polynomial goes zero, so it follows that our polynomial $f \left(x\right)$, is

$f \left(x\right) = {\left(x + 2\right)}^{2} \left(x - 3\right) \left(x + 4 i\right)$

Because that meets both of our requirements: being the product of four monomials and going to zero when all we use a value of $x$ that is a zero of the function.

If you wish you can multiply it back, as it can be more useful that way sometimes (like differentiation or integration) but if you want to graph it, or look for roots, or divide some other function by it, you'll probably like it better this way.