Partial Fractions — Proper and Improper Rational Fractions

In business calculations we often work with a rational fraction — one polynomial divided by another, such as $\dfrac{3x+5}{(x-1)(x+2)}$. Before such an expression can be split into simpler pieces, it must first be classified.

A rational fraction $\dfrac{f(x)}{g(x)}$ (with $g(x) \neq 0$) is called:

• Proper, if the degree of the numerator $f(x)$ is strictly less than the degree of the denominator $g(x)$. Example: $\dfrac{2x+3}{x^2-1}$ (degree 1 over degree 2).
• Improper, if the degree of the numerator is greater than or equal to the degree of the denominator. Example: $\dfrac{x^3+1}{x^2-4}$ (degree 3 over degree 2).

Why the distinction matters: the partial-fraction method covered in the next section applies directly only to a proper fraction. An improper fraction must first be reduced by ordinary polynomial long division:

where $Q(x)$ is the quotient (a polynomial) and $r(x)$ is the remainder, with $\deg r(x) < \deg g(x)$ — so $\dfrac{r(x)}{g(x)}$ is now a proper fraction that the partial-fraction method can act on.

This distinction is not just a technicality — it shows up naturally in business contexts. An average-cost expression such as $\dfrac{\text{Total Cost}(x)}{x}$, or a rational demand/revenue expression whose numerator's degree has grown through algebraic manipulation, is often improper and must be reduced by division before it can be analysed term by term.