Question

Hello:) For partial fractions, is
Quotient = remainder / diviser
a proper fraction?

Answer 1

Answer:

You can represent most things with a partial fraction, anything in the form:

$\frac{f(x)}{(x+\alpha )(x-\beta )}=\frac{A}{(x+\alpha )}+\frac{B}{(x- \beta )}$

Notice how the roots of the equation from the denominator on the left hand side are used to form the denominations of fractions on the right hand side.

We can now multiply both sides by the denominator on the left hand side.

$f(x)=A(x-\beta )+B(x+ \alpha )$

We can now setup 2 simultaneous equations by substituting $x$ for your values of $-(- \beta )$ and $-\alpha$ to make $A$ or $B$ 0, allowing you to find the other $A$ or $B$ value.

$f(-\alpha ) = A(-\alpha - \beta) + 0\\f(\beta ) = 0 + B(\beta + \alpha )$