We have polynomials f(x),g(x) (degree>1), their GCD has degree>1.

I have to prove that there exist polynomials u(x),v(x) such that:

f(x).u(x)=g(x).v(x) and deg(u)<deg(g),deg(v)<deg(f).$\displaystyle a^-1(x)$

Am I right?

let d(x) be their GCD, then we have:

f(x)=a(x).d(x)

g(x)=b(x).d(x)

so $\displaystyle f(x).a^-1(x)=g(x).b^-1(x)$ where and $\displaystyle b^-1(x)$ are polynomials u(x),v(x)

is the proof of existence ok?

I need help with showing the inequality about degrees. Can anybody help me please?

Thank you for your help.