Why do you need two fractions for partial fraction decomposition with repeated factors?

Question

For example, suppose my denominator contains $(x - 1)(x - 1)$.
I know I need two fractions, one with $(x - 1)$ and one with $(x - 1)^2$ as the denominator. But I'm looking for a deeper reason as to why. It makes sense when you go through and get a common denominator that it will all work out in the end, but I just want a good explanation for it.

Answer 1

Source:

Consider a fraction in which the denominator is $8$. Does that mean that the denominator of each term being added together had to be an $8$? No, the denominators could have been $2, 4$, or $8$ because the common denominator between $2, 4$, and $8$ is $8$. The implications of this for partial fraction decomposition are that when you have a repeated factor (a factor with a multiplicity other than one), you need to include a factor in the expansion for each power possible. For example, if you have an $(x-2)^3$, you will need to include an $(x-2)$, an $(x-2)^2$, and an $(x-2)^3$. The exponents of $2$ or $3$ does not change whether the factor is linear or quadratic, only how many times the factor is there. Each of those $(x-2)$ factors would receive a constant term in the numerator because $x-2$ is linear, no matter what power it is raised to.

Hope it helps.

Answer 2

You don't actually need two fractions for the squared factor. If for example you wanted to express $\frac{4x}{(x+1)(x-1)^2}$ in partial fractions, you could express it as $$\frac{4x}{(x+1)(x-1)^2}\equiv\frac{a}{x+1}+\frac{bx+c}{(x-1)^2}$$ This gives you $$4x\equiv a(x-1)^2+(bx+c)(x+1)\equiv (a+b)x^2+(-2a+b+c)x+(a+c)$$and you can then solve the equations:

1)$\ \ a+b=0$

2)$\ -2a+b+c=4$

3)$\ a+c=0$

for $a,b$ and $c$ to get$$\frac{4x}{(x+1)(x-1)^2}\equiv \frac{-1}{x+1}+\frac{x+1}{(x-1)^2}$$The point is that you have three unknowns ($a,b$and $c$) and three equations, which will generally give a solution.

Were you to assume$$\frac{4x}{(x+1)(x-1)^2}\equiv\frac{a}{x+1}+\frac{b}{(x-1)^2}$$in this case you would get$$4x\equiv a(x-1)^2+b(x+1)\equiv ax^2+(-2a+b)x+(a+b)$$ You then have to solve

1)$\ a=0$

2)$\ -2a+b=4$

3)$\ a+b=0$

This is three equations in two variables and has no solution.

Trying $$\frac{4x}{(x+1)(x-1)^2}\equiv\frac{a}{x+1}+\frac{b}{(x-1)^2}+c$$ would not help because then after clearing the denominator you finish up with a cubic, so you still have one more equation than you have variables.

Assuming $$\frac{4x}{(x+1)(x-1)^2}\equiv\frac{a}{x+1}+\frac{b}{(x-1)}+\frac{c}{(x-1)^2}$$is usually the most convenient.

If you want to split a rational function into a sum of reciprocals of polynomials then this last form is necessary, but only possible if the denominator of the function you wish to split (in its lowest terms) is a product of linear factors.