Polynomial questions use the factor theorem, the remainder theorem, division and sketching. The reports show that most of the marks lost are slips, in signs, in the roots, and in what the question asks you to state. 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
- Substituting the wrong value. Using x = 4 instead of x = −4, or substituting −a instead of a.
- Not explaining the factor theorem. Showing that x + 4 is a factor without saying that if x = −4 is a root then x + 4 is a factor, or stating only that p(−4) = 0 without showing the calculation.
- Division errors. Errors in the algebra when finding the quotient, sign slips during the division, and, for synthetic division, not allowing for the extra factor of 2 that arises from the leading coefficient. Reports note that the remainder theorem is easier than long division when only a remainder is needed. Some students carried forward an incorrect initial factor into the next part.
- Setting up equations for unknowns. Equating an expression to 12 or 0 instead of −12, losing the −12 from the polynomial, sign errors in terms involving b, and misreading −6 as 6. Reports suggest checking the coefficient of x³ against the factors you found. A few students found the correct a and b from an incorrect process, such as thinking that subtracting −4a + b = −8 from 9a − b = 27 would eliminate b, and did not earn full credit.
- Non-integer answers. When the question says that a and b are integers, a non-integer answer is a sign of a small error. Reports advise checking the work to find it.
- Missing roots. Omitting one of the roots (such as x = 1), forgetting the negative square root (omitting x = −0.5), or not stating the solutions after factorising. One report describes a student who factorised 4x² − 1 incorrectly as (4x − 1)(4x + 1).
- Roots and factors. Some students did not understand the difference between roots and factors, and offered one for the other.
- Comments on solutions. In a question about real solutions, the comment should say "real solutions", not just "solutions".
- Sketching a cubic. Responses often lacked attention to detail. Errors included placing the minimum point on the y-axis, not labelling the y-intercept (or finding it incorrectly), drawing a negative cubic where a positive one was needed, and drawing a quadratic. Another error was stating all the y-coordinates as 0.
- "Hence" with a graph. Where the question says "hence", the graph should be used to obtain the inequalities, with no calculations needed.
What examiners advise
- Show the substitution and the working for p(−a), and state the factor theorem conclusion in words.
- Check signs during division and when forming equations.
- State the solutions of the equation after factorising, including negative roots.
- Label the intercepts on a sketch and check the shape for the sign of the leading coefficient.
- Use the hint in the question (integer values, the shape of the curve) to check the answer.
Habits that help
- After division, multiply the factors back out to check you get the original polynomial.
- When a question gives a list of integers, test your answer against it before moving on.
- Write "roots" and "factors" in different columns when you list them.
Questions students ask
How do I use the factor theorem properly?
Show the substitution and the working for p(−a), and state the conclusion in words: if x = −4 is a root then x + 4 is a factor. Students substituted x = 4 instead of x = −4, or stated only p(−4) = 0 without showing the calculation.
What is the difference between a root and a factor?
A root is a value of x, such as −4, and a factor is an expression, such as x + 4. Reports note some students offered one for the other.
How do I make sure I do not miss roots?
State the solutions after factorising, including negative roots such as x = −0.5. Reports record x = 1 omitted and negative square roots forgotten.
How do I sketch a cubic?
Label the y-intercept and the x-intercepts, check the shape against the sign of the leading coefficient, and do not place the minimum point on the y-axis. Where the question says "hence", use the graph to get inequalities with no calculation.
What if the question says a and b are integers?
Use it as a check. A non-integer answer is a sign of a small error, so go back and find it.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.