Different methods of Partial Fractions.

Question

By the method of partial fractions we take:

$$\frac{px+q}{\left(x-a\right)\left(x-b\right)}=\frac{A}{x-a}+\frac{B}{x-b}$$ $$\frac{px+q}{\left(x-a\right)^2}=\frac{A}{x-a}+\frac{B}{\left(x-a\right)^2}$$

For both kinds of fractions. I don't understand this part, what do we actually do when we split a fraction into partial fractions. Why don't we write:

$$\frac{px+q}{\left(x-a\right)^2}=\frac{A}{x-a}+\frac{B}{\left(x-a\right)}$$ (like we do in the first part).

Answer 1

Note

$$\begin{align}\frac{A}{x-a}+\frac{B}{x-a}=\frac{A+B}{x-a}\end{align}$$

^ Taking a glance at this, should make it clear

If it doesn't:

$$\begin{align}\frac{A+B}{x-a} \neq \frac{px+q}{(x-a)^2}\end{align}$$

You get that the denominator isn't equal, and also $A+B$ is a constant whereas $px+q$ isnt.

Answer 2

Maybe we can pretend having no knowledge about what the result should be

Then for the first one, we want to write $px + q$ in the form of $A(x-b) + B(x-a)$, such that we can have $$\dfrac{px+q}{(x-a)(x-b)} = \dfrac{A(x-b) + B(x-a)}{(x-a)(x-b)} = \cdots$$

For the second one, we can see it's not useful to write $px+q$ in the above form, we'd rather prefer writing it in form of $A(x-a) + B$, such that we can have

$$\dfrac{px+q}{(x-a)^2} = \dfrac{A(x-a) + B}{(x-a)^2} = \cdots$$