How do you express (x²+2) / (x+3) in partial fractions?

1 Answer
Feb 22, 2016

x/1 + {-3x+2}/{x+3}

Explanation:

because the top quadratic and the bottom is linear you're looking for something or the form

A/1 + B/(x+3), were A and B will both be linear functions of x (like 2x+4 or similar).

We know one bottom must be one because x+3 is linear.

We're starting with
A/1 + B/(x+3).
We then apply standard fraction addition rules. We need to get then to a common base.

This is just like numerical fractions 1/3+1/4=3/12+4/12=7/12.

A/1 + B/(x+3)=> {A*(x+3)}/{1*(x+3)} + B/(x+3)={A*(x+3)+B}/{x+3}.
So we get the bottom automatically.

Now we set A*(x+3)+B=x^2+2
Ax + 3A + B=x^2+2
A and B are linear terms so the x^2 must come from Ax.
let Ax=x^2 => A=x
Then
3A + B=2
substituting A=x, gives
3x + B=2
or
B=2-3x
in standard from this is B=-3x+2.
Putting it all together we have
x/1 + {-3x+2}/{x+3}