- #1
RJLiberator
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Homework Statement
If [itex]gcd(f(x),g(x)) = 1[/itex] and m,n ∈ ℕ, show that [itex]gcd(f(x)^m, g(x)^n) = 1[/itex].
Homework Equations
The Attempt at a Solution
So I had previously proved this for non-polynomials:
gcd(a,b)=1
then gcd(a^n,b^n)=1
Proof: a = p1*p2*...*pn
b = p1*p2*...*pm
then
a^n = p1^n*p2^n*...*pn^n
b^n = p1^n*p2^n*...*pm^n
Since a and a^n have the same prime factors and b and b^n have the same prime factors and a and b are relatively prime then a^n and b^n are relatively prime.
I guess, I am looking at this question from the same point of view.
Is there a difference here with polynomials? Why would there be?