I'm reading the theorem, but I don't quite understand it. It says:
If f(x) is an element of $Z[x]$ then f(x) factors into a product of two polynomials of lower degrees $r$ and $s$ in $Q[x]$ if and only if it has such a factorization with polynomials of the same degrees r and s in $Z[x]$
I don't quite understand what it means. Can someone explain it to me without using really advanced math talk. I ain't that quite advanced in my mathematics. If you can do it with as a simple example as possible, that would be appreciated.
It means that if you have a polynomial, like $3x^2+x-2$, whose coefficients are all integers (in this case, $3, 1, -2$) and it factors with rational coefficients
e.g. $(x-\frac{2}{3})(3x+3)$, where the coefficients of the two factors are $1, -\frac{2}{3}$ and $3 ,3$
then there is also a factorization where all the coefficients are integers $(3x-2)(x+1)$.