Irreducible quadratic factors; partial fraction decomposition.

Question

Please help me understand why there is Dx+E, Fx+G etc, instead of the regular A's, B's, C's etc. What is it about the irreducible quadratic in the denominator that makes it different on top?

Answer 1

First of all, recall that for $\dfrac{p(x)}{q(x)}$, the degree of $p(x)$ has to be less than $q(x)$ in order for us to apply partial fraction decomposition. It then follows that since the denominator has irreducible quadratic factors, the numerator can either have a constant term $D$ or it's also possible that the numerator has a linear term $Ex+F$. This is because the degree of the denominator is $2$, so the degree of the numerator can either be of degree $0$ (constant term) or degree $1$ (linear term).