Polynomial/Bezout/X^2+X+1 and X-1/Example

We want to determine the greatest common divisor for the polynomials ${\displaystyle {}X^{2}+X+1}$ and ${\displaystyle {}X-1}$ in ${\displaystyle {}\mathbb {Q} [X]}$. For this, we perform the division with remainder, and we obtain

${\displaystyle {}X^{2}+X+1={\left(X+2\right)}{\left(X-1\right)}+3\,.}$

Thus the two polynomials are coprime. A representation of the ${\displaystyle {}1}$ is

${\displaystyle {}1={\frac {1}{3}}{\left(X^{2}+X+1\right)}-{\frac {1}{3}}{\left(X+2\right)}{\left(X-1\right)}\,.}$