Question

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

Answer 1

See below.

Explanation:

First, distribute:

`=(1)/(x(x+1))`

Now, perform partial fraction decomposition. This is only two factors, so we only need linear functions:

`(1)/(x(x+1)) =(A)/(x)+(B)/(x+1)`

Now, multiply everything by x(x+1):

`1=A(x+1)+B(x)`

Now, solve for each variable. Let's let x=-1:

`1=A(-1+1)+B(-1)`

`1=0A-B=-B`

`B=-1`

Now, let's let x=0:

`1=A(0+1)+B(0)`

`A=1`

So:

`(1)/(x(x+1))=(1)/(x)-(1)/((x+1))`

Double Check:

`(1)/(x)-(1)/((x+1))=((x+1))/(x(x+1))-(x)/(x(x+1))`

`=(x-x+1)/(x(x+1)) =(1)/(x^2+x)`