What is the formula of converting fraction into addition of fractions?

Question

What is the formula of converting fraction into addition of fractions? $$\frac{4-s}{(s-3)(s-2)(s+1)}=\frac 1 4 \frac 1 {s-3} - \frac 2 3 \frac 1 {s-2} +\frac 5 {12} \frac 1 {s+1}$$

Answer 1

Partial Fraction Decomposition

$$\frac{4-s}{(s-3)(s-2)(s+1)} = \frac{A}{s-3} + \frac{B}{s-2} + \frac{C}{s+1}$$

$$\therefore 4 -s = A(s-2)(s+1) + B(s-3)(s+1) + C(s-3)(s-2)$$

To find the values of $A, \space B, \space C$, simply let $s$ equal certain values $0, \space 1, \space 2, ...$ (choosing specific values can make it extremely easy) and solve the ensuing equations.

Answer 2

This is called a partial fraction decomposition.