Can Rational Functions be Written as a Sum of Polynomials?

[silvermane]

Homework Statement

Take two polynomials f(x) and g(x) over a field, and suppose that gcd(f,g)=1.
Show that any rational function, $$\frac{h(x)}{f(x)g(x)}$$ can be written in the form
$$(\frac{p(x)}{f(x)})$$ + $$(\frac{q(x)}{g(x)})$$ for some polynomials, p and q.

The Attempt at a Solution

I claim that any rational function $$(\frac{h(x)}{f(x)g(x)})$$ can be written in the form $$(\frac{p(x)}{f(x)})$$ + $$(\frac{q(x)}{g(x)})$$ for some polynomials, p and q.

We're given that the gcd(f,g) = 1, so we could write them as a linear combination of their gcd:
1 = u(x)f(x) + w(x)g(x).
We divide both sides by f(x) and g(x) to obtain:
$$(\frac{1}{f(x)g(x)})$$ = $$(\frac{u(x)}{g(x)})$$ + $$(\frac{w(x)}{f(x)})$$
We could then multiply both sides by h(x) to obtain:
$$(\frac{h(x)}{f(x)g(x)})$$ = $$(\frac{h(x)*u(x)}{g(x)})$$ + $$(\frac{h(x)*w(x)}{f(x)})$$

Now, let p(x) = h(x)*u(x) and q(x) = h(x)*w(x). Thus, we have that
$$(\frac{h(x)}{f(x)g(x)})$$ = $$(\frac{p(x)}{g(x)})$$ + $$(\frac{p(x)}{f(x)})$$
and we've proven our claim.

I just want to make sure that what I've done here is correct. Just so that I understand the material and what's going on. Thank you for your help in advance!

 

[micromass]

seems fine.
It's a cool proof actually...

 

[silvermane]

Thanks! I had a lot of fun writing it once I figured the whole thing out.
Thanks for checking it for me!