Partial Fraction Decomposition—Multiple Variables

[END]

What's the best approach to solving the partial-fraction decomposition of the following expression?

$$\frac{1}{(a-y)(b-y)}$$

The expression is not of the following forms:

But I know the solution is

$$= \frac{1}{(a-b)(y-a)}-\frac{1}{(a-b)(y-b)}$$

 

[ShayanJ]

I don't use such tables. Any time I want to do a partial fraction decomposition, I just write (e.g.) \frac{1}{(a-y)(b-y)}=\frac{A}{a-y}+\frac{B}{b-y} and then determine A and B.
Anyway, if you multiply the factors you'll see that its in fact in the form of the third entry in the table!

 

[Mark44]

What the table is saying is the for each distinct (i.e., not repeated) factor (ax + b) in the denominator, you'll have a term $\frac{A}{ax + b}$ in the decomposition. So $\frac{1}{(a - y)(b - y)}$ results in $\frac{A}{a - y} + \frac{B}{b - y}$.

Equate the two expressions and solve for A and B, which is more or less what Shyan said.