Partial fractions with irreducible denominators above degree 2

Question

In all courses and textbooks I have seen covering partial fractions, they only cover cases where the denominator is reducible to linear or quadratic factors. However, I can not see any reason why having an irreducible factor of degree greater than 2 in the denominator would present a problem to the decomposition. Obviously, this could be factored into its complex roots and done the normal way, but if, for example, we took $$\frac{1}{(1+x+x^2+x^3+x^4)(1+x+x^2)}=\frac{Ax^3+Bx^2+Cx+D}{1+x+x^2+x^3+x^4}+\frac{Ex+F}{1+x+x^2}$$ would this pose any theoretical problems?

Answer 1

Why not just decompose it as something with a 6th degree polynomial in the denominator? Then I could tell you right off that the numerator is $1$ !

The point of partial fractions (as an integration technique) is that the things you end up with on the right hand side are all easy (relatively) to integrate: you get logs, a few inverse trig functions, some polynomial terms, and some messy stuff for the linear/(power of a quadratic) bits.

The first expression on the right-hand side of your expression isn't easily integrable, so it's not great for doing integration.

CAN you express the LHS as a sum of the form on the RHS? Sure. Would you do so when trying to integrate? Probably not.

As others observe, the term "irreducible" in your question is misplaced, but even without that, I figured it was worth answering.