When you add algebraic fractions you put them over a common denominator and end up with a single, more complicated fraction. Partial fractions run that machine in reverse: they take one fraction whose denominator is a product of factors and split it back into a sum of simpler fractions.

Suppose the bottom factorises into two distinct linear pieces. Then we can always write

for some constants A and B. Each piece on the right has just one factor underneath — much friendlier to work with. The whole task is to find those two numbers.

Finding the constants leans on the factor theorem's idea of substituting a clever value of x: choose the value that makes one factor vanish, and that term drops out, leaving a single unknown. (When more than one constant survives at once, you are really just solving by substitution.) This is the bread and butter of integrating rational functions, where breaking one awkward fraction into simple pieces makes each piece easy to handle.

See it split

Step through the decomposition of \frac{5x + 1}{(x + 1)(x - 2)}. We propose the split, clear the denominators, then substitute the two values of x that knock out one unknown at a time.

See it explained

Sal Khan works a partial fraction expansion from scratch, finding the unknown constants.