1 Proper and improper fractions
Frequently we find that an algebraic fraction appears in the form
$\phantom{\rule{2em}{0ex}}\text{algebraicfraction}=\frac{\text{numerator}}{\text{denominator}}$
where both numerator and denominator are polynomials. For example
$\phantom{\rule{2em}{0ex}}\frac{{x}^{3}+{x}^{2}+3x+7}{{x}^{2}+1},\phantom{\rule{2em}{0ex}}\frac{3{x}^{2}2x+5}{{x}^{2}7x+2},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{x}{{x}^{4}+1},$
The degree of the numerator, $n$ say, is the highest power occurring in the numerator. The degree of the denominator, $d$ say, is the highest power occurring in the denominator. If $d>n$ the fraction is said to be proper ; the third expression above is such an example. If $d\le n$ the fraction is said to be improper ; the first and second expressions above are examples of this type. Before calculating the partial fractions of an algebraic fraction it is important to decide whether the fraction is proper or improper.
Task!
For each of the following fractions state the degree of the numerator ( $=n$ ) and the degree of the denominator ( $=d$ ). Hence classify the fractions as proper or improper.
$\text{(a)}\phantom{\rule{1em}{0ex}}\frac{{x}^{3}+{x}^{2}+3x+7}{{x}^{2}+1},\phantom{\rule{2em}{0ex}}\text{(b)}\phantom{\rule{1em}{0ex}}\frac{3{x}^{2}2x+5}{{x}^{2}7x+2},\phantom{\rule{2em}{0ex}}\text{(c)}\phantom{\rule{1em}{0ex}}\frac{x}{{x}^{4}+1},\phantom{\rule{2em}{0ex}}\text{(d)}\phantom{\rule{1em}{0ex}}\frac{{s}^{2}+4s+5}{\left({s}^{2}+2s+4\right)\left(s+3\right)}$

Find the degree of denominator and numerator and hence classify (a):
The degree of the numerator, $n$ , is 3. The degree of the denominator, $d$ , is 2.
Because $d\le n$ the fraction is improper.

Here
$n=2$
and
$d=2$
. State whether (b) is proper or improper:
$d\le n$ ; the fraction is improper.

Noting that
$x={x}^{1}$
, state whether (c) is proper or improper:
$d>n$ ; the fraction is proper.

Find the degree of the numerator and denominator of (d):
Removing the brackets in the denominator we see that $d=3$ . As $n=2$ this fraction is proper.
Exercise
For each fraction state the degrees of the numerator and denominator, and hence determine which are proper and which are improper.
 $\frac{x+1}{x}$ ,
 $\frac{{x}^{2}}{{x}^{3}x}$ ,
 $\frac{\left(x1\right)\left(x2\right)\left(x3\right)}{x5}$
 $n=1,d=1$ , improper,
 $n=2,d=3$ , proper,
 $n=3,d=1$ , improper.
The denominator of an algebraic fraction can often be factorised into a product of linear and/or quadratic factors. Before we can separate algebraic fractions into simpler (partial) fractions we need to completely factorise the denominators into linear and quadratic factors. Linear factors are those of the form $ax+b$ ; for example $2x+7$ , $3x2$ and $4x$ . Irreducible quadratic factors are those of the form $a{x}^{2}+bx+c$ such as ${x}^{2}+x+1$ , and $4{x}^{2}2x+3$ , which cannot be factorised into linear factors (these are quadratics with complex roots).