Why, how and when do we use partial fractions and polynomial long division?

Partial fractions and long division have the same purpose which is to decompose complex expressions into simpler ones in order to easily deal with them in calculations. Think of them as tools that we use for 2 different types of fractions. We use partial fractions to simplify proper fractions(the degree of the numerator is higher than the degree of the denominator) while polynomial long division is used for improper fractions(the degree of the numerator is smaller than the degree of the denominator).Steps for partial fractions: factorise the denominator; for factors with exponents, you need one partial fraction per exponent from 1 up; when forming a partial fraction, the numerator needs to be one degree less than the denominator e.g. for quadratics you need to put Ax+ B in the numerator. Good question to let the student work with: (x^2 + 15)/(((x+3)^2)*(x^2+3)) as it requires the student to apply all the concepts. When it comes to polynomial long division, it is easier to demonstrate it on the white board.