Page:Calculus Made Easy.pdf/142

way he can add together three or more fractions. Now this process can certainly be reversed: that is to say, that if this last expression were given, it is certain that it can somehow be split back again into its original components or partial fractions. Only we do not know in every case that may be presented to us how we can so split it. In order to find this out we shall consider a simple case at first. But it is important to bear in mind that all which follows applies only to what are called “proper” algebraic fractions, meaning fractions like the above, which have the numerator of a lesser degree than the denominator; that is, those in which the highest index of $x$ is less in the numerator than in the denominator. If we have to deal with such an expression as ${\dfrac {x^{2}+2}{x^{2}-1}}$, we can simplify it by division, since it is equivalent to $1+{\dfrac {3}{x^{2}-1}}$; and ${\dfrac {3}{x^{2}-1}}$ is a proper algebraic fraction to which the operation of splitting into partial fractions can be applied, as explained hereafter.

Case I. If we perform many additions of two or more fractions the denominators of which contain only terms in $x$, and no terms in $x^{2}$, $x^{3}$, or any other powers of $x$, we always find that the denominator of the final resulting fraction is the product of the denominators of the fractions which were added to form the result. It follows that by factorizing the denominator of this final fraction, we can find every one of the denominators of the partial fractions of which we are in search.