So I'm confused about the partial fraction decomposition of fractions that involved repeated factors in the denominator. From what I've been taught, when you are breaking your fraction into partial fractions, the numerator of each fraction has a degree of at least one less than the denominator. However, from what I've seen in textbooks and online the partial fraction decomposition of fraction involving repeated factors in denominators often looks like this for linear factors. The degree of the numerator is only one less than the degree of the repeated factor which is $(x-a)$ rather than than the degree of the entire denominator
$\frac{P(x)}{(x - a)^n} = \frac{A}{(x - a)} + \frac{B}{(x -a)^2} + \frac{C}{(x -a)^3} + \dots$
But if I consider the general rule I was taught. Then shouldn't each partial fraction look like this.
$\frac{P(x)}{(x - a)^n} = \frac{A}{(x - a)} + \frac{Bx+C}{(x -a)^2} + \frac{Dx^2+Ex+F}{(x -a)^3} + \dots$
Why is this so? Is there a proof or explanation for this? because I can't seem to find one online. I've found one proof from Cambridge Mathematics 4 unit Year 12, but I don't understand the proof. So could you explain their proof as well, https://i.gyazo.com/ad52aefdf68e4728985498b48262a1ed.png (the proof) especially the part where it express $P(x)$ as a polynomial in $(x-a)$, what does that mean? and what are they doing when they say let $G(x) = P(x+a)= b_{n-1}x^{n-1} + b_{n-2}x^{n-2} + \dots$