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silvermane
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Homework Statement
We have two polynomials F(x) and g(x) over a field, and suppose that we know gcd(f,g)=1.
Consider two rational functions b(x)/f(x) and c(x)/g(x) in which we have deg(b)<deg(f) and deg(c)<deg(g). Prove that b(x)/f(x) = c(x)/g(x) can only be true when b(x) = c(x) = 0
The Attempt at a Solution
I really need help starting this proof.
So far, I have that f(x) won't divide b(x) since deg(b)<deg(f), and g(x) won't divide c(x) since deg(c)<deg(g). Because of this, we cross multiply to obtain:
b(x)*g(x) = c(x)*f(x)
But then it gets hazy here.
I know that f(x) and g(x) won't cancel out since their gcd=1, and b(x) must equal c(x) for them to cancel out. But then we'll be left with g(x) = f(x), so b(x) must equal c(x) and both terms must be equal to 0.
Is that all I have to do/say? It seems too simple.
Thanks for your help in advance! It means a lot!