Partial Fraction Decomposition: Proof of General Case with Linear Factors
In this video I go over an extensive proof of decomposing rational functions for the general case with linear factors. Although the video is very long and detailed, it follows the same style as in my earlier proof on the simplest case in which a simple theorem was first proved in which it was repeatedly applied to the overall partial fraction decomposition proof. This is a very extensive and long video but it is worthwhile following along to get an idea of just how much detail is required in proving many of the math theorems out there.
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Partial Fraction Decomposition: General Case for Linear Factors Proof
In my last video I went over the proof for decomposing rational functions for the simplest case of having proper fractions with linear and non-repeating factors such as the following examples:
What if the factors repeat? Such as:
The general case with linear factors:
where:
- ai's are all unique
- N(x) is a polynomial of degree at most n1 + n2 + … + nd - 1
- i.e. forms a proper fraction where degree of numerator is less than degree of denominator
To prove this we will first prove a similar form of this theorem:
Lemma 2:
Let N(x) and D(x) be polynomials of degree n and d respectively, with n < d + m. Suppose that a is NOT a zero of D(x), i.e. D(a) ≠ 0. Then there is a polynomial P(x) of degree p < d and numbers A1, …. , Am such that:
Proof of Lemma 2:
To prove Lemma 2 we need to be able to find the polynomial P(x) and numbers A1, …. , Am.
We first multiply both sides by D(x)(x - a)m :
Thus we have shown that there are numbers A1, … , Am and polynomial P(x) with degree of p < d (i.e. at most d - 1) such that:
Now back to the previous theorem:
We can apply Lemma 2 but with:
