How to separate an equation into partial fractions?

Question

I am looking at a math question that has simplified this:

into this:

Can somebody explain the process for how this simplification was made? i.e. how does the denominator get broken down to those two terms?

Answer 1

Here's a quick and easy way for simple or basic fractions$$\frac 1{y(1-y)}=\frac {\color{blue}{1-y}+y}{y(\color{blue}{1-y})}=\frac {1-y}{y(1-y)}+\frac y{y(1-y)}=\frac 1y+\frac 1{1-y}\color{brown}{=\frac 1y-\frac 1{y-1}}$$

Answer 2

We write $$ \frac {1}{y(1-y)}=\frac {A}{y}+\frac {B}{1-y}$$

Upon taking common denominator , we get $$\frac {A(1-y)+By}{Y(1-y)}=\frac {1}{y(1-y)}$$

Therefore we want to have,$$ A(1-y)+B(y)=1$$

Let $y=1$ and we get $B=1$ and let $y=0$ to get $A=1$

Thus $$ \frac {1}{y(1-y)}=\frac {1}{y}+\frac {1}{1-y}=\frac {1}{y}-\frac {1} {y-1} $$