In this video I go over further into partial fraction decomposition and show how to decompose rational functions that have non-linear factors (or irreducible factors) in the denominator. The method used for this case is similar to the technique used for linear factors but we have to account for the "size" of the numerator in the partial fractions so we need to apply write the numerator of each non-linear factor as a polynomial.

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Partial Fraction Decomposition: Non-Linear Factors

In my earlier videos I went over techniques in partial fraction decomposition for the cases where the factors were linear, such as:

What if the factors were non-linear (irreducible)? Consider the following rational function:

Now in this case if we were to apply the same technique as previously shown, we come across a problem:

The problem is that the B coefficient is paired with a x to the power of 1 while the A coefficient is paired with a x to the power of 2. This means that one of the numerators of the partial fractions is being forced to have a higher power than the other one.

Thus we need to make both coefficients be paired with the higher power. I will detail the proof of this in a later video so stay tuned!

Basically accounting for the "size" of the numerator we modify the normal technique for partial fraction decomposition: