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

Polynomial questions in Paper 3 centre on division, the factor and remainder theorems, finding unknown constants, and solving cubic or quartic equations. The method is well known to most students. The marks go missing in the layout of the division, in sign slips, and in what is substituted. 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

• Untidy long division. Reports say that unclear setting out leads to columns that are not aligned, and then to subtraction errors. Many students made sign errors when completing the process, with final-step mistakes such as 2 − (−3) = 6 or ±1.
• The missing term. When a polynomial has no x² term, a zero must be written in its place. Students who did not allow for the missing term got incorrect quotients and remainders.
• Arithmetic slips in the division. A considerable number of students did not obtain the constant term of the quotient correctly or made an error in a term of the remainder.
• Substituting the wrong value. Using x = 3 when the factor was x + 2, or using f(−3) = 0 when the root was not −3. The root of a factor ax + b is −b/a.
• Treating 2x + 3 as a factor when it was not, or mixing up the factor and the remainder.
• The remainder. After dividing, some students did not know what to do with the remainder, for example in an expression of the form (b − 2)x + a − 7, which has to be set equal to zero or compared with a given form. Others did not realise they needed to equate real and imaginary parts when complex roots were involved.
• Complex roots found wrongly. When a quadratic factor had complex roots, some students gave real roots such as −1 and 2 instead. One report notes that the first mark, for stating that the square roots of −3 are solutions, should have been accessible to all. Others assumed the quadratic factor must have real factors.
• Wrong setup. Students who started with the wrong setup often scored nothing other than the first mark. Some students divided the wrong way round, or stopped after the first term of the quotient.
• Not showing enough. For the factor and remainder theorems, show the substitution and the evaluated value, for example f(3) = 0 written out in full. Where the question asks you to justify a result, an answer without justification loses marks.
• Using the calculator for roots. The rubric states that no marks are given for unsupported answers from a calculator, and one report notes that finding roots in that way should have been avoided.

What examiners advise

• Line up like terms in a long division, in a vertical column.
• Think about which method is quicker. Reports note that students who expanded quadratic brackets and compared coefficients were generally more successful than those who attempted algebraic long division. Long division is valid, but the arithmetic is error-prone.
• Show the substitution in the factor and remainder theorems.
• Use the remainder correctly: the quotient plus the remainder over the divisor if you are asked to rewrite a fraction.
• Clear fractions or choose signs sensibly to make the algebra easier.

Habits that help

• Write 0x² explicitly when a power is missing.
• Check the division by multiplying the quotient by the divisor and adding the remainder.
• Check by substituting a simple value of x (such as 1) into both sides.

Questions students ask

Why does my polynomial long division go wrong?

Often because of untidy setting out. Align like terms in vertical columns, and write a zero for a missing term, such as 0x² when there is no x² term. Check the division by multiplying the quotient by the divisor and adding the remainder.

What value do I substitute in the factor theorem?

The root of the factor ax + b, which is −b/a. Students used x = 3 when the factor was x + 2, or treated 2x + 3 as a factor when it was not. Show the substitution and the evaluated value, such as f(3) = 0 written out in full.

What do I do with the remainder?

It depends on the question. If it is in the form (b − 2)x + a − 7, set it equal to zero or compare it with the given form. If you rewrite a fraction, use the quotient plus the remainder over the divisor. With complex roots, equate real and imaginary parts.

Is it better to use long division or to compare coefficients?

Reports note that students who expanded the brackets and compared coefficients were generally more successful. Long division is valid, but its arithmetic is error-prone.

Can I use my calculator to find the roots of a cubic?

Not for marks. The rubric says unsupported calculator answers earn no marks, and one report notes that finding roots that way should have been avoided. Show the factor theorem and the division.