Let f = X⁷ - 1 and g = X⁵ - 1 in R[X]. The polynomials a and b in af + bg = gcd(f,g) obtained by the extended Euclidean algorithm are:

X³ + X + 1    and    X² - X²

-X³ - X    and    X⁵ + X³ + 1

-2X³    and    X⁵ - X³