# Partial Fractions

## Partial Fractions

If f(x) and g(x) are two polynomials, then $$\frac{f\left( x \right)}{g\left( x \right)}$$ defines a rational algebraic function or a rational function of x.

If degree of f(x) < degree of g(x), then $$\frac{f\left( x \right)}{g\left( x \right)}$$ is called an proper rational function.

If degree of f(x) ≥ degree of g(x) then $$\frac{f\left( x \right)}{g\left( x \right)}$$ is called an improper rational function.

If $$\frac{f\left( x \right)}{g\left( x \right)}$$is an improver rational function. We divided f(x) by g(x) so that the rational function. $$\frac{f\left( x \right)}{g\left( x \right)}$$ is expressed I the $$Q\left( x \right)+\frac{f\left( x \right)}{g\left( x \right)}$$ where Q(x) and f(x) are polynomial such that degree of f(x) is less than that of g(x) . thus $$\frac{f\left( x \right)}{g\left( x \right)}$$ is expressed  as the sum of a polynomial and a proper rational function.

any proper rational function $$\frac{f\left( x \right)}{g\left( x \right)}$$ can be expressed as the sum of rational functions, each having a simple factor ofg(x) . Each such fraction is called a partial fraction and the process of obtaining them is called the resolution or decomposition of $$\frac{f\left( x \right)}{g\left( x \right)}$$ into partial fractions.

Case I: when denominator is expressed is expressible as the product of non-repeating linear factors.

Let   g(x) = (x – a₁) (x – a₂) … (x – an) then, we assume that $$\frac{f\left( x \right)}{g\left( x \right)}=\frac{{{A}_{1}}}{x-{{a}_{1}}}+\frac{{{A}_{2}}}{x-{{a}_{2}}}+….+\frac{{{A}_{n}}}{x-{{a}_{n}}}$$.

Where, A₁ , A₂……are constants and can be determined by equating the numerator on RHS to the numerator on LHS and then substituting  x= a₁ , a₂ ….. an.