Binomial expansion in Additional Math 0606: common mistakes and examiner advice

Binomial expansion questions in Additional Math ask for terms of expansions such as (1 + ax)ⁿ or (2x − 3/x)ⁿ, a coefficient or constant term, or unknowns found from given coefficients. The reports show that most errors are in the powers and the brackets, not in the binomial theorem itself. 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

• Powers applied to only part of the term. The most common error in several questions: not applying the power to the "a" as well as the "x" in the third and fourth terms, forgetting any powers of 2 in the coefficients, or using b instead of b² when finding c. One report says writing (2x²) without brackets meant it was unclear whether the power applied to the 2 as well as the x².
• Squares and signs. Not squaring the 3 in (−1/(3y))², not keeping the negative sign right after squaring, and using (px)² incorrectly or expanding it wrongly.
• Slips in a term. For example, simplifying ⁵C₂(ax)² as 10ax² (leaving out the a²), and replacing aⁿ⁻¹ with b⁴⁻¹ in a general term.
• Not all the terms. Missing one or two of the terms needed to find a coefficient (for example, missing 20a⁴), or not using enough terms from the answer to the previous part.
• Unknowns. Offering only positive values for a and b in an otherwise correct solution, rejecting or not using the negative value of a, finding a = ±2 and then miscalculating b or c when substituting a = −2, and reversing the values of b and c. A small number did not state the values of b and c explicitly.
• Not writing the expansion. Some candidates did not state the expansions they were using and then chose incorrect strategies. Others multiplied out (a + x)(a + x)(a + x)(a + x) by hand.
• The equation for the coefficients. The equation should be formed using the coefficients, without the x². In other questions, students tried to find an equation in terms of n instead of a, or mixed up the equations, and some recovered while others did not.
• Combination and factorial notation. Not being able to express the combination as a fraction involving factorials, simplifying (59 − (r + 1))! incorrectly to (60 − r)!, or trying to solve the equation with x included.
• Manipulating the 7. Incorrect manipulation of the 7, with (7 + 7x)ⁿ also commonly stated or used.
• Ascending powers. Writing "ascending powers of x" does not mean that the constant 1 should be rewritten in that form.
• Using "hence." Not following the instruction to use part (a), and giving a partially correct solution without a final answer that does not involve x.
• Showing steps. The general rubric says all steps should be shown, so stating the correct solution without at least a correct quadratic was not accepted as a valid method.
• Presentation. Poor presentation led to miscopying errors or other slips in working. Some candidates miscopied the question.

What examiners advise

• Put brackets around every term in the expansion, such as (2x²)³ and (−1/(3y))², and apply the power to the number, the letter, and the sign.
• Write out the expansion you are using, with the binomial coefficients, before simplifying.
• Use "hence" and the result from part (a), and check you have all the terms that contribute.
• Refer to the list of formulas on page 2 for the binomial theorem.
• State your final values clearly, and in the order asked for.

Habits that help

• Write each term of the expansion on its own line, then simplify one line at a time.
• To check an expansion, substitute a simple value such as x = 1 into the original and into your answer.
• For a constant term, work out which term gives x⁰ before you write any coefficients.

Questions students ask

Why does (ax)² become a²x² and not ax²?

Because the power applies to the number, the letter and the sign. It is the most common error in several questions. Put brackets around every term in the expansion, such as (2x²)³ and (−1/(3y))².

Do I need to write out the expansion?

Yes, write the expansion you are using, with the binomial coefficients, before simplifying. The general rubric says all steps should be shown, and a correct solution stated without at least a correct quadratic was not accepted.

How do I find unknown values from given coefficients?

Form the equation using the coefficients, without the x². Offer both positive and negative values of a and b where they arise, and state your final values in the order asked for.

What does "ascending powers of x" mean?

The terms go from the lowest power of x to the highest, starting with the constant term. It does not mean the constant 1 should be rewritten in a different form.

What does "hence" mean in a binomial question?

Use the result of the previous part. Check you have all the terms that contribute, and finish with an answer that does not involve x if the question requires it.