What is the multiplicity of the root 1 of $X^{2 n}-n X^{n+1}+n X^{n}-X^{2}$ for $n \in \mathbb{N}_{\geq 2}$?
1 is at least a root of multiplicity 1. We derive the polynomial and obtain: $2 n X^{2 n-1}-n(n+1) X^{n}+n^{2} X^{n-1}-2 X$. 1 is a root if and only if $n=2$. We derive once more if $n=2$: $12 X^{2}-12 X+4$ for which 1 is not a root. Therefore, 1 is always a simple root except if $n=2$ and then $c'$ is a double root.

Another way to determine the existence of a root of a polynomial is to use continuity: a continuous function is such that if $x$ approaches $a$, then $f(x)$ will be close to $f(a)$ for all $x$ and $a$. A polynomial satisfies the intermediate value property: if $a_{1}<a_{2}$ are real numbers and $P\left(a_{1}\right)$ and $P\left(a_{2}\right)$ have different signs, then $P$ has a root in $\left[a_{1} ; a_{2}\right]$.

Suppose that $P$ is of degree $d$ and leading coefficient $a$. When $x$ tends to $\pm \infty, P(x)$ will behave the same way as $a \cdot x^{d}$ (the ratio of $P(x)$ and $a \cdot x^{d}$ approaches 1 indefinitely as $|x|$ becomes very large).