1. Polynomials

Assistance and help on this would be great! thank you

Let F be a field. Let a(x), b(x) exist in F[x] be polynomials such that a(x)≠ 0 or b(x) ≠ 0.
Let d(x)=gcd(a(x), b(x), that is, d(x) is the monic polynomial in F[x] of highest degree such
that d(x)|a(x) and d(x)| b(x). Suppose that d1(x) exist in F[x] is a monic polynomial such that
d1(x)=a(x)u(x) + b(x)v(x) for some u(x), v(x) that exist in F[x], d1(x)| a(x) and d1(x)| b(x)
Prove that d(x) = d1(x).

2. What results do you know in connection with this? See if you can follow the following argument with your current knowledge.

Since $d_1(x) | a(x)$ and $d_1(x) | b(x)$, and since $d(x)$ is the greatest divisor of both $a(x)$ and $b(x)$, we have $d_1(x) | d(x)$.

Conversely, since $d(x)$ divides both $a(x)$ and $b(x)$, we have that $d(x)$ divides $a(x)u(x)+b(x)v(x)$, i.e. $d(x) | d_1(x)$. In conclusion, $d(x) = \lambda d_1(x)$ for some scalar $\lambda$. Since both $d(x)$ and $d_1(x)$ are monic, the scalar is equal to 1, hence $d(x)=d_1(x)$.