Perform the Euclidean division of $2 X^{3}+5 X^{2}+6 X+1$ by $X^{2}-3 X+2$.

## Roots of a Polynomial

We notice that an expression in factored form, such as $a_{n}\left(X-r_{1}\right) \cdots\left(X-r_{n}\right)$, is also a polynomial (we will see later how to relate the $r_{i}$ to the coefficients $a_{i}$ of the polynomial). It has the advantage of being very manageable: we can identify the points where it vanishes, which are then the $r_{i}$.

Definition 2.24. A real root of a polynomial $P$ is a real number $r$ such that $P(r)=0$. Their search is almost obsessive when studying a polynomial.

Remark 2.25. In the following statements, we will exclude the case of the zero polynomial, for which every real number is a root.

Let us note the following fundamental theorem:

Theorem 2.26. Let $P$ be a polynomial of degree $n$. It has exactly $n$ complex roots (not necessarily distinct), and therefore at most $n$ real roots.

The proof is beyond the scope of our discussion; the interest lies in bounding the number of possible roots. Sometimes, we can also bound it from below:
With the notations from the course: $S=2 X+11$ and $R=35 X-21$.