4 Proper fractions with quadratic factors

Sometimes when a denominator is factorised it produces a quadratic term which cannot be factorised into linear factors. One such quadratic factor is x 2 + x + 1 . This factor produces a partial fraction of the form A x + B x 2 + x + 1 . In general a quadratic factor of the form a x 2 + b x + c produces a single partial fraction of the form A x + B a x 2 + b x + c .

Key Point 16

A quadratic factor a x 2 + b x + c in the denominator produces a partial fraction of the form

A x + B a x 2 + b x + c
Task!

Express as partial fractions 3 x + 1 ( x 2 + x + 10 ) ( x − 1 )

Note that the quadratic factor cannot be factorised further. We have

3 x + 1 ( x 2 + x + 10 ) ( x − 1 ) = A x + B x 2 + x + 10 + C x − 1

 First multiply both sides by ( x 2 + x + 10 ) ( x − 1 ) :

( A x + B ) ( x − 1 ) + C ( x 2 + x + 10 )

Evaluate C by letting x = 1 :

4 = 12 C so that C = 1 3

Equate coefficients of x 2 and hence find A , and then substitute any other value for x (or equate coefficients of x ) to find B :

− 1 3 , 7 3 . Finally express in partial fractions:

3 x + 1 ( x 2 + x + 10 ) ( x − 1 ) = − 1 3 x + 7 3 x 2 + x + 10 + 1 3 x − 1 = 7 − x 3 ( x 2 + x + 10 ) + 1 3 ( x − 1 )