Representing a Fraction as a Sum of Fractions (Splitting Fractions)
Big Idea: Any single fraction — especially algebraic fractions — can be split (decomposed) into a sum of simpler fractions. This technique is essential in algebra, calculus, and beyond.
1. The Core Principle
You already know that we can add fractions:
$\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$
Splitting fractions is simply the reverse process:
$$\dfrac{a + b}{c} = \dfrac{a}{c} + \dfrac{b}{c}$$
This works because division distributes over addition. If the numerator is a sum (or difference), we can split the fraction into separate fractions, each sharing the same denominator.