Remember for partial fractions when you get a quadratic factor that cannot be expressed as linear factors you have you use
So for the first one we start off with
Now that
will result in the form I originally spoke about above, and the other x term will just give us the normal result.
So we are looking at
Which then gives
Now if we equate the coefficients we get
C=3, A=1
A+B=0
B=-A
B=-1
So that gives
The second one can be approached in the same method.
Now for the last one, because the greatest power of the numerator is equal to or greater than the greatest power of the denominator, you will have to long divide first, then apply the methods for partial fractions. So for the numerator the greatest power is
and for the denominator the greatest power will be
, so we have to long divide first.
So that brings us to
Then from there just apply the normal rules again.
Then add the other x back in later.