<s>[INST] Why the linear numerator for fractions with irreducible denominators and constant numerators for reducible denominators? For example:
$\Large{\frac{2x^3+5x+1}{(x^2+4)(x^2+x+2)}}$ breaks down to $\Large{\frac{ax+b}{x^2+4}+\frac{cx+d}{x^2+x+2}}$
I have been told that since the denominators are irreducible, the numerators will be either linear or constant.
Now my question is for something like $\Large{\frac{2x^3+5x+1}{x^2-4}}$ you would make it equal $\Large{\frac{A}{x+2}+\frac{B}{x-2}}$, why do we assume that the numerators are constant? Why couldn't they be linear??
 [/INST] Because remainder of division is always smaller then the degree of the polynomial so we assume it's exactly 1 smaller,and find if the other terms are zero
$$\frac{2x^3+5x+1}{(x+2)(x-2)}=\frac{ax+b}{x+2}+\frac{cx+d}{x-2}=\frac{ax+b-a(x+2)}{x+2}+a+\frac{cx+d-c(x-2)}{x-2}+c=\frac{ax+b-ax-2a}{x+2}+\frac{cx+d-cx+2c}{x-2}+a+c=\frac{b-2a}{x+2}+\frac{2c+d}{x-2}+a+c$$
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