Solve the equation $\cos 2 x+3 \cos x=1$.
Let $y=\cos x$, then the equation becomes $2 y^{2}+3 y-2=0$, so $y=\frac{-3 \pm 5}{4}$. Since $-1 \leqslant y \leqslant 1$, we necessarily have $y=1 / 2$, thus $x= \pm \frac{\pi}{3}+2 k \pi$ with $k \in \mathbb{Z}$.

## Use of Euclidean Division

There are formulas to solve equations of degree 3 and 4, but not of higher degree.

However, we can look for rational roots, then factor the expression.

Proposition 119. Let $a_{0}, \ldots, a_{n}$ be integers, such that $a_{0}$ and $a_{n}$ are non-zero. If $p$ and $q$ are integers that are coprime, and if $x=\frac{p}{q}$ is a rational root of $a_{0}+\cdots+a_{n} x^{n}$, then $p$ divides $a_{0}$ and $q$ divides $a_{n}$.

Indeed, $a_{0} q^{n}+\cdots+a_{n} p^{n}=0$, so $p$ divides $a_{1} q^{n-1} p+\cdots+a_{n} p^{n}=-a_{0} q^{n}$. Since $p$ is coprime with $q$, it divides $a_{0}$. We reason similarly for the other assertion.