>> Back in August 2010 I posted a .pdf file of a handout I wrote >> for a high school precalculus class that I taught several >> times in the mid to late 1990s. However, I used the result >> to prove various irrationality results, which I thought was >> a more mathematically important and interesting application. >> Moreover, coming up with the polynomials to apply the rational >> root theorem to helped students practice algebraic manipulation >> skills, which I also thought was a more mathematically important >> skill than using the theorem to factor contrived polynomials. >> >> You can find the .pdf file at
Given your interest in this topic, here are some more items you might be interested in. At the beginning of the integration handouts (first item below) I give several "high school" ways to factor 1 + x^4, 1 + x^5, and 1 + x^6. The sci.math post "Integrating Rational Functions" also has some relevant aspects, such as a proof (assuming the Fundamental Theorem of Algebra) that every polynomial with real coefficients can be factored into polynomials with real coefficients whose degrees are at most 2.