Let $f(x) = x^4 + x^3 + x + 1$ and $\,g(x) = x + 1$ be two polynomials in $\mathbb{Z_2[x]}$
Find $d(x) = \gcd(f(x), g(x))$ and write $d(x)$ as a polynomial combination of both $f(x)$ and $\:g(x)$.
My attempt:
Dividing $f(x)$ by $g(x)$ using the division algorithm, we get $f(x) = (x+1)(x^3+1)$, and we have $g(x) = x + 1$.
Thus $d(x) = x+1$ ($\gcd(f(x), g(x)$) and $d(x) = 1.g(x) + 0.f(x)$
Is my attempt correct? Thank you