# Partial Fraction Decomposition—Multiple Variables

1. Dec 4, 2014

### 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

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

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2. Dec 4, 2014

### 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!

3. Dec 5, 2014

### Staff: Mentor

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.