The algebraic expressions of the types (px+q)/(x-a)(x-b), px2+qr+r/(x-a2)(x-b1), px2+qr+r/(x-a)(a2+bx+c), etc. are called rational functions in which both numerator and denominator are called polynomial functions. Let f(x) and g(x) are two polynomials, then defines a rational algebraic functions, or a rational function of x. If the degree of f(x) < the degree of g(x), then is called a proper rational function. If thedegree of f(x) the degree of g(x), then is called an improper rational function.
Any proper rational function f(x)/g(x) can be expressed as the sum of rational functions, each having a simple factor of g(x). Each such fraction is called a partial fraction, and the process of obtaining them is called the resolution, or decomposition of f(x)/g(x) into partial fractions.
The resolution of f(x)/g(x) into partial fractions depends mainly upon the nature of the factors of g(x) as discussed below.
TYPE -1, When denominator (g(x)) is expressed in the form of g(x) =(x –a1) (x – a2) (x – a3) …… then, we can transform it into a sum of simple fractions i.e.
f(x)/g(x) = A/(x-a1) + A/(x-a2) + A/(x-a3) +……..
Where, A, B, C…… are constants, and can be determined by equating the numerator on R.H.S. to the numerator on LHS, and then substituting x = a1+ a2+a3……..an.