Example 3 (Continued)

We have

One might recognize a³ + 3x² + 3x + 1 at once as (x + 1)³. Alternatively, one can see easily that a = -1 is a root of x³ + 3x² + 3x + 1. Therefore x + 1 is a factor of it. Dividing by x + 1 we get the quotient x² + 2x + 1 = (a + 1)².

The sum of partial fractions has the form

Then

Solving these three equations for A, B, and C we have A = 1, B = -2, C = 1.

Therefore