# What is the relationship between the degree of a polynomial and the maximum numbers of zeros it can have?

This theorem tells us that any polynomial $P \in \mathbb{C} \left[X\right]$ has EXACTLY $n$ roots in $\mathbb{C}$ (each one of them is different or not), if $n = \mathrm{de} g \left(P\right)$.
This means that you can write $P$ as the product of simple polynomials in $\mathbb{C}$ : $P \left(X\right) = {\left(X - {a}_{1}\right)}^{{\alpha}_{1}} {\left(X - {a}_{2}\right)}^{{\alpha}_{2}} \ldots {\left(X - {a}_{n}\right)}^{{\alpha}_{n}}$ with ${a}_{i}$ defined such that $P \left({a}_{i}\right) = 0$ and ${\sum}_{i = 1}^{n} {\alpha}_{i} = n = \mathrm{de} g \left(P\right)$.