Partial fractions in A Level Math Paper 3: common mistakes and examiner advice

Partial fractions rarely stand alone in Paper 3. They are the first part of a question that continues into an integration or a binomial expansion. The reports show that a wrong start here costs marks all the way to the end of the question. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• The wrong form. In one question with a repeated factor in the denominator, the majority of students started with a form that did not match it (two simple fractions only) and could score a maximum of 2 marks. With a repeated factor the form must include the squared denominator. When the fraction is improper (the numerator has the same or a higher power than the denominator), it must be divided first. Students who did not divide, or chose the wrong form, found the question highly challenging.
• The method fails in different ways. Reports describe that an error in the comparison method often made all of the unknowns wrong, whereas an error in the substitution method often left one of the unknowns correct. Arithmetic slips also appear in solving the resulting equations, for example 2B = 3 + 3 giving B = 6.
• Wrong coefficients in the expansion. After partial fractions, students expand each term using the binomial series. The mistakes are in the factors taken out: (3 − x)⁻¹ should become 3⁻¹(1 − x/3)⁻¹, but students extracted 3 and 9 instead of 3⁻¹ and 3⁻², lost a factor, used the wrong sign, or did not evaluate the squared term correctly. Writing (x − 1)⁻² as −(1 − x)⁻² was also common.
• Combining expansions. Errors when combining the expansions with Ax + B and C, lost factors of 2, and sign errors in the x³ term.
• Integrating the result. A term of the form (Ax + B)/(x² + 2) was often not split. Students ignored the numerator, assumed the integral would be of the form p tan⁻¹ q, or knew it was a logarithm but retained x as a coefficient. The coefficient of the ln term was often wrong, as was the coefficient inside the ln (such as ln(2x − a) and −ln(3a − x)).
• Missing brackets. In expressions like (x + 2)/(x + 2), omitted brackets cost marks.
• Reversing the working. When an answer is given and the integral does not match it, many students tried to adjust their incorrect result to reach it. One report says their time would be better spent reviewing the solution to find the error.
• Radians. Reports remind students to use radians when working with trigonometric functions in calculus problems.

What examiners advise

• Check the form before you start: is the fraction proper? Is there a repeated factor? Is there a quadratic factor that cannot be factorised?
• Check your partial fractions quickly by recombining the terms.
• Write out the expansion fully, with the factor taken out written as a power of a number: 3⁻¹, 3⁻², 2⁻¹.
• Show enough detail in the integration, since correct final answers from incorrect coefficients earn little without working.
• Look at the numerator to see which standard integral is needed.

Habits that help

• Always check the degrees of the numerator and the denominator first.
• Use a value of x that makes one factor zero, and compare coefficients for the rest.
• Substitute a simple value of x into the original fraction and into your partial fractions to see whether they agree.

Questions students ask

How do I choose the right form for partial fractions?

Check first: is the fraction proper, is there a repeated factor, is there a quadratic factor that cannot be factorised? In one question with a repeated factor, most students started with two simple fractions and could score a maximum of 2 marks.

Which method is safer, substitution or comparing coefficients?

Reports say an error in the comparison method often made all the unknowns wrong, while an error in the substitution method often left one correct. Whichever you use, check by recombining the terms.

How do I expand partial fractions with the binomial series?

Take the constant out so the bracket starts with 1. (3 − x)⁻¹ should become 3⁻¹(1 − x/3)⁻¹. Write the factor as a power of a number, such as 3⁻¹, 3⁻² or 2⁻¹, and show the unsimplified terms first.

How do I integrate a term like (Ax + B)/(x² + 2)?

Split it. Students ignored the numerator, assumed the answer was p tan⁻¹ q, or knew it was a logarithm but kept x as a coefficient. Look at the numerator to see which standard integral is needed.

Why does a mistake in partial fractions cost so many marks?

Because they are usually the first part of a question that continues into an integration or a binomial expansion. A wrong start costs marks all the way to the end.