Let's say you have to spread a cubic equation into partial fractions. What you would normally get is a linear factor and a quadratic or a linear and 2 linear expressions. My question is, for that quadratic factor ,if the solutions are fractions (for example : $ (x + \frac{1}{2})(x+\frac{2}{3}) $), then are you supposed to break it down to $ \frac{A}{x + \frac{1}{2}} + \frac{B}{x+\frac{2}{3}} $ ? What i want to know is : if solutions are not whole numbers , do you have to reduce the factor, and why not or why?
And what if the discriminant is not a perfect square and i end up with surds while factorizing? Do i then still split it into 2 factors?
If you want to perform integration, you have to perform the partial decomposition even if the roots are not integers.
Why? Because this is necessary to take antiderivatives.