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

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

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

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

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