Equal-degree fractions before partial fractions
Partial fractions can split only a proper remainder. Compare the degrees before writing any letters.
Equal degrees require one division step
Compare degrees first
numerator degree lower
go straight to partial fractions
degrees equal
divide first → constant + proper fraction
numerator degree higher
outside the current partial-fractions requirement
- If the numerator degree is lower, the fraction is already proper.
- If the degrees are equal, divide out a constant and split the proper remainder.
- The current syllabus excludes partial-fraction questions where the numerator degree is greater than the denominator degree.
Key idea
Equal degrees do not mean “partial fractions is impossible”. They mean the answer begins with a constant.
See the constant before splitting the remainder
Worked example
Current-paper five-mark decomposition
Factor the denominator and divide the leading terms. The constant quotient is .
Substituting gives ; substituting gives .
Answer
(9709/32/M/J/24 Q2)
Recombine to expose a missing term or sign
Put the answer over the original denominator. Its numerator must return exactly to the given numerator. This is faster than redoing the decomposition when a sign looks doubtful.
Common mistake
Do not begin with only when the original degrees are equal. Two proper pieces cannot create the missing constant quotient.
Your turnIndependent check [5 marks]
Express in partial fractions.
Show worked answer
Answer