Binomial expansion in A Level Math Paper 1: common mistakes and examiner advice

Binomial expansion is one of the more predictable topics in Paper 1, and one of the most carefully marked for slips. The questions are usually short. Almost every lost mark in this topic comes from the same few causes, which are listed below. 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 of a product. Writing (ax)² as ax² (or (ax)³ as ax³) instead of a²x² (or a³x³). One report says this mistake is seen every year in this paper. The same slip appears with a negative term: (−2q)² or (2x²)² is mishandled.
• Signs. Minus signs are omitted, or both terms become positive or both negative. Reports also mention confusing ascending with descending order.
• Forgetting to raise the coefficient to the power. Examples are using 15 × 3x² instead of 15 × (3x)², forgetting to cube the 2, or forgetting to square or cube a number from the second bracket.
• Selecting the wrong term. Students stopped before reaching the required term, found only the term in x³ when x² was also needed, or misread a request for "the terms as far as x²" and gave only the term in x².
• Not noticing x⁰. In products of two expansions, students forgot that 1 is a term in x⁰ and omitted it from the products.
• Combining terms. When two products contribute to the same power of x, students equated the two terms instead of adding them, or missed one of the pairs.
• "Independent of x" misunderstood. This means the constant term, and a few answers included x (such as −160x).
• Equation solving after expansion. Students solved for p, but made sign errors or did not show clearly how they solved the resulting quadratic. Others missed the negative solution when taking a square root.
• Miscopying. Errors in binomial coefficients came from miscopying the powers they had written down.

What examiners advise

• Use the general term to find the required term, rather than writing out the whole expansion. Reports describe writing out all terms, or multiplying out five brackets, as time-consuming and a source of many errors.
• Pre-select the products you need. In a product of two expansions, list which pairs of terms give the power of x you want before multiplying.
• Use brackets around the whole term, including its sign and its coefficient, when raising to a power.
• Be careful when simplifying (ax)ⁿ.
• Use the answer from an earlier part when the question says to.
• Give the answer in the form requested.

Habits that help

• Write the general term with every part in brackets before substituting numbers.
• Check the power of x on your chosen term before working out the coefficient.
• Check signs once more at the end.

Questions students ask

Why does (2x)² become 4x², not 2x²?

Because the whole bracket is squared, including the number. Writing (ax)² as ax² is the mistake one examiner report says is seen every year in this paper. Put brackets around the whole term, including its sign and coefficient, when you raise it to a power.

Do I need to write out the whole binomial expansion?

No. Examiners advise using the general term to find the required term. Reports describe writing out all the terms, or multiplying out five brackets, as time-consuming and a source of many errors.

How do I find a coefficient in a product of two expansions?

List which pairs of terms give the power of x you want before multiplying, and add the products. Students forget that 1 is a term in x⁰, equate two contributions instead of adding them, or miss one of the pairs.

What does "independent of x" mean?

It means the constant term, the one with no x. A few answers included x, such as −160x.

How do I avoid sign mistakes in a binomial expansion?

Keep the sign inside the bracket with the term, and check the signs once more at the end. Reports record minus signs being omitted, both terms becoming positive or both negative, and ascending order confused with descending.