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

In Paper 3, the binomial expansion is for negative and fractional powers and is valid only for a certain range of x. It is often combined with partial fractions. The reports show that the method is well known and the same small slips cost marks every session. 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

• Not taking out the constant correctly. This is the most repeated mistake. To expand (3 − x)⁻¹ you take out 3 and write 3⁻¹(1 − x/3)⁻¹. An incorrect statement such as (9 − 3x)^(1/2) = 9(1 − 3x)^(1/2) was common. Other students extracted 3 and 9 instead of 3⁻¹ and 3⁻², put the coefficient in the numerator when it belonged in the denominator, or lost the constant altogether.
• Constants left unsimplified. 2⁻¹ and 3⁻¹ often became 2 and 3. In a combined question, B and C were combined with 2⁻¹ or 2⁻² incorrectly.
• Using x when the bracket contains something else. The most common error was to expand with powers of x, rather than powers of (−2x) or (6x), so the coefficients were wrong. Another version was using x in place of x/2, or 3x replaced by x.
• Wrong index. An incorrect power was used for the expansion, or a negative sign was lost from the index.
• Arithmetic. Many students gained the two method marks but then made arithmetic errors, such as 215 instead of 217, or lost some factors of a in the denominators.
• Order of terms. Terms were written in descending instead of ascending order, or the correct answer was multiplied by 3 to remove fractions when the form asked for was different.
• Range of validity. Some students could not identify the correct range of values of x, with incorrect answers such as |x| < ½ and x > −2.
• Partial fractions. Where the expansion follows a partial-fractions step, the mistakes in that step carry over (see the partial fractions guide). A significant number of students gave no response to the final part of one question.
• No working. One report notes that when no working is shown and the process goes wrong, no credit can be given for having a correct method.

What examiners advise

• Take out the constant first, so that the bracket starts with 1, and write the constant as a power before you simplify it.
• Expand with the whole term in brackets, such as (−2x)² and (−2x)³.
• State the range of validity, using the term inside the bracket, as an inequality for |x|.
• Check your partial fractions by recombining before you expand.
• Show the unsimplified terms first, then simplify. The first marks are often for the unsimplified form.

Habits that help

• Write the general expansion of (1 + u)ⁿ at the side, and substitute u as a whole.
• Check the constant term and the x term against a quick numerical estimate.
• Check that the answer is in ascending powers of x.

Questions students ask

How do I expand (3 − x)⁻¹ using the binomial series?

Take out the constant first so the bracket starts with 1: 3⁻¹(1 − x/3)⁻¹. An incorrect statement such as (9 − 3x)^(1/2) = 9(1 − 3x)^(1/2) was common.

Why are my coefficients wrong when the bracket has −2x or 6x in it?

Because the expansion is in powers of the whole term, not x. Write (−2x)² and (−2x)³ in brackets. The most common error was expanding with powers of x instead of (−2x) or (6x).

How do I give the range of validity?

As an inequality for |x| using the term inside the bracket. Reports record students unable to identify it, with wrong answers such as |x| < ½ and x > −2.

What order should the terms be in?

Ascending powers of x. Terms written in descending order, or multiplied by 3 to clear fractions when a different form was asked for, lose marks.

Why do I lose accuracy even when my method is right?

Arithmetic slips after the method marks, such as 215 instead of 217, 2⁻¹ and 3⁻¹ becoming 2 and 3, and lost factors in denominators. Show the unsimplified terms first, then simplify.