11. Express

as a sum of rational functions with linear denominators.

First factor the denominator.

Then this fraction can be obtained by adding two fractions of the form

because if we find common denominators,

we get

If we set this equal to our original expression

a sufficient condition for these to be equal would be for

and

This gives us a system of two equations in two unknowns to solve. If we subtract the top equation from the bottom one we get

so

If we substitute this into the top equation we get

Subtracting 1 from both sides gives us

so we have

To check, we add.

Find common denominators

as required.