Let F be a field and let a(x), b(x), and c(x) be polynomials in F[x] with gcd (a(x),b(x))=1 and with a(x)b(x)=c(x)^n for some n>1. Prove: n=2, a(x)=p₁(x)², and b(x)=q₁(x)².
Let F be a field and let a(x), b(x), and c(x) be polynomials in F[x] with gcd (a(x),b(x))=1 and with a(x)b(x)=c(x)^n for some n>1. Prove: n=2, a(x)=p₁(x)², and b(x)=q₁(x)².
who said $\displaystyle n$ must be 2? for example put $\displaystyle a(x)=x^n, \ b(x)=(x+1)^n,$ where $\displaystyle n$ is whatever you like. there must be something missing in your problem.