In these works it is proved that any rational fraction may be resolved into a series of partial fractions; and that
(1) To any factor of the first degree, as x-a, in the denominator there corresponds a partial fraction of the form {A}{x-a}.
(2) To any factor of the first degree, as « — b, occurring n times in the denominator there corresponds a series of n partial fractions of the form,
{B}{x-b} + {C}{x-b}^2 + + {R}{x-b}^n
(8) To any quadratic factors, as x^2+ px + q, in the denominator there corresponds a partial fraction of the form
{Ax + B}{x^2+ px + q}
(4) To any quadratic factor, as x^2+ px + q, occurring n times in the denominator there corresponds a series of n partial fractions of the form
{Ax + B} {(x^2+ px + q)}+ {Cx + D} {(x^2+ px + q)^2}+ + {Rx+s}{(x^2+ px + q)^n}.
Here the quantities A, B, C, D,--- R, S, are all independent of a.
We shall make use of these results in the examples that follow.
Ex. 1. Separate {5x-11}{2x^2 +x-6} into partial fractions.
Since the denominator 2x^2 +x-6= (x+2)(2x-3), we assume
{5x-11}{2x^2 +x-6} = {A}{x+2} + {B}{2x-3}
where A and B are quantities independent of x whose values have to be determined.
Clearing of fractions,
5x-11=A(2x-3)+ B(x+2).
Since this equation is identically true, we may equate coefficients of like powers of x; thus.
2A+B=5, -3A+2B=-11; whence A= 3, B= -1. {5x-11}{2x^2 +x-6}={3} {x+2} - {1} {2x-3}
