Rational polynomials I'm not very familiar with algebra and was wondering if there are any results regarding the effective order of rational polynomials (i.e. rational functions). 
Specifically: given $P(z)$ and $Q(z)$ as polynomials in $z$ with real coefficients of order $p$ and $q$ respectively, is there a way to know the effective order of the rational function $P(z)Q^{-1}(z)$  - by which I mean to know if there are $r$ roots shared by the numerator and denominator that cancel each other out resulting in a smaller rational polynomial of order $p-r, q-r$? 
More importantly, can this be done  without explicit factorization ? I just want to know the value of $r$ without knowing the roots or the reduced polynomials.
 A: You can determine whether two polynomials have common roots by calculating a resultant. A resultant is a function of the coefficients of the two polynomials that is zero iff they have a common root. Resultants are related to "elimination theory", which was a hot topic 100 years ago, but became unfashionable in the 1930s. 
The Wikipedia and MathWorld pages talk about "the" resultant of two polynomials. But, in fact there are numerous different types of resultant. They are associated with the names of people like Cayley, Sylvester, Dixon, and Bezout, among others.
See also this question.
A: Euclid's GCD algorithm can be used to find common factors between two polynomials.  For instance:
$$
\begin{align}
\gcd(x^3 + 1, x^4 + x^2 + 1) & = \gcd(x^3 + 1, x^4 + x^2 + 1 - x(x^3 + 1))
\\ & = \gcd(x^3 + 1, x^2 - x + 1)
\\ & = \gcd(x^3 + 1 - x(x^2 - x + 1), x^2 - x + 1)
\\ & = \gcd(x^2 - x + 1, x^2 - x + 1)
\\ &= \gcd(x^2 - x + 1, 0)
\\ &= x^2 - x + 1
\end{align}
$$
So $x^2 - x + 1$ is the greatest common factor between $x^3 + 1$ and $x^4 + x^2 + 1$.
