Proving Equality of Rational Functions with Two Polynomials

[silvermane]

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!

 

[jbunniii]

b(x)*g(x) = c(x)*f(x)

g(x) divides the LHS so it must divide the RHS. Since gcd(g,f) = 1, this means g(x) must divide c(x). By the same reasoning, f(x) must divide b(x). But you have some constraints on the degrees of these polynomials. What do the constraints imply?

 

[jbunniii]

By the way, the question may not make any sense as worded. If b(x) is the zero polynomial, what is deg(b)? Some people would say deg(b) is undefined in that case, others would set it to -\infty. What is the definition in this context?

 

[silvermane]

g(x) divides the LHS so it must divide the RHS. Since gcd(g,f) = 1, this means g(x) must divide c(x). By the same reasoning, f(x) must divide b(x). But you have some constraints on the degrees of these polynomials. What do the constraints imply?

The constraints imply that f(x) can't divide b(x) and g(x) can't divide c(x). So then we would have that they must be 0 for the two sides to be equal?

 

[jbunniii]

The constraints imply that f(x) can't divide b(x) and g(x) can't divide c(x). So then we would have that they must be 0 for the two sides to be equal?

Or, putting it more pedantically, f(x) cannot divide a polynomial of lower degree unless that polynomial is zero. [This assumes "degree" is defined for the zero polynomial.] Therefore b(x) must be the zero polynomial, and similarly for c(x).

 

[silvermane]

Or, putting it more pedantically, f(x) cannot divide a polynomial of lower degree unless that polynomial is zero. [This assumes "degree" is defined for the zero polynomial.] Therefore b(x) must be the zero polynomial, and similarly for c(x).

Ah, that makes sense. Thanks so much for your help