What is an improper fraction, and how to I make thisproper so that it can be differentiated?

An improper fraction is any fraction where the order of the constant (power of x) is equal or greater in the nuberator than the denominator, take for example;

(x²-3x-2)/(x²-3x+2)

This can first be factorised to give (x²-3x-2)/(x²-3x+2)=(x²-3x-2)/(x-1)(x-2).

We then set this fraction to be equal to a series of constants, divided by the factors i.e

(x²-3x-2)/(x-1)(x-2) = A/(x-1) + B/(x-2).  Multiplying by the denominator gives us;

(x²-3x-2) = A(x-2) + B(x-1).  This can be solved for A and B by substitution of  values.  The best place to start is by setting one factor equal to 0, in this case we first let x=2 so that the A term is 0, and find B=-4.  We then set x=1 so that the B term=0 to find A=4.  We can then write:

(x²-3x-2)/(x²-3x+2) = 4/(x-1) -4/(x-2).  Which is an easier form to handle for further calculation such as differentiation.