Sometimes it is useful to express a single fraction such as the sum of 2 (or more in other cases) separate fractions.

This is called decomposing a function, f(x), in partial fractions.

Consider this example:

This equation can be split into the sum of two single fractions.

Therefore

To find the value of the constants A and B depends on the factors in the denominator. Follow the examples below.

Using the example above:

Adding the two fractions on the right hand side gives:

As the denominators are now the same the numerators must match as well. Therefore:

If we choose to substitute x = -1, this will eliminate the constant B. This gives:

If we choose to substitute x = -2/3 to this will eliminate the constant A. This gives:

Therefore:

Lets take the example:

As this has a quadratic factor in the denominator we need to write the partial equations in the following format:

Once again we need to add the two fractions on the right hand side. This gives:

Once again the denominators are now the same, so the numerators must be the same as well. Therefore:

By substituting x = 0, we eliminate B and C. This gives:

We can substitute two other values for x and solve B and C by simultaneous equations.

Consider this example:

The denominator contains (x − 2)². Since, (x − 2)² = (x − 2 )(x − 2) it is called a repeated factor. When there is a repeated factor in the denominator we need to write the partial equations in the following format:

Once again we add the fractions on the right hand side together and thus the denominators are the same, so the numerators are the same, giving:

We again proceed by substitution to obtain the values of the constants.