6 Improper fractions
When calculating the partial fractions of improper fractions an extra polynomial is added to any partial fractions that would normally arise. The added polynomial has degree where is the degree of the numerator and is the degree of the denominator. Recall that
a polynomial of degree 0 is a constant, say,
a polynomial of degree 1 has the form ,
a polynomial of degree 2 has the form ,
and so on.
If, for example, the improper fraction is such that the numerator has degree 5 and the denominator has degree 3, then , and we need to add a polynomial of the form .
Key Point 17
If a fraction is improper an additional term is included taking the form of a polynomial of degree , where is the degree of the numerator and is the degree of the denominator.
Example 42
Express as partial fractions
Solution
The fraction is improper because , and so . Here , so we need to include as an extra term a polynomial of the form , in addition to the usual partial fractions. The linear term in the denominator gives rise to a partial fraction . So altogther we have
Multiplying both sides by we find
Equating coefficients of gives .
Equating coefficients of gives and so .
Equating the constant terms gives and so .
Finally, we have
Exercise
Express each of the following improper fractions in terms of partial fractions.
- ,
- ,
- ,
- ,
- ’
- ,
- ,
- ,