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.