About this chapter
The Pure 4 binomial expansion works for any power, including negative and fractional ones, but only for small enough x. So every expansion now comes with a range of validity, and many questions use one to approximate a root or a reciprocal, then ask how accurate the estimate is.
This chapter covers the general expansion, replacing x with a bracketed term such as −4x, taking out a power of the constant, validity, unknown constants in an expansion, products of brackets, expanding partial fractions, approximations and percentage errors. One question models the distance to the horizon from a height above the Earth.
The ten questions
- Q1 (5 marks): the first four terms of 1/√(9 + 4x) with exact fractional coefficients, and where it is valid.
- Q2 (6 marks): expand the cube root of (1 − 6x), then match a product with it to find a, b and c.
- Q3 (6 marks): two given coefficients of an expansion of (1 + kx) to a power n give k and n.
- Q4 (6 marks): partial fractions first, then the series expansion and its range of validity.
- Q5 (6 marks): expand (4 − x) to the power −½, then substitute x = 1 to approximate √3 as a fraction.
- Q6 (11 marks): combine two expansions to approximate √((1+x)/(1−x)), then a rational estimate of √6 and its percentage error.
- Q7 (11 marks): three partial fractions with a repeated factor, their expansion, and the general coefficient of the nth power of x.
- Q8 (10 marks): an expansion of (2 − kx) to the power −3 with a condition linking two coefficients, then an approximation.
- Q9 (10 marks): the distance to the horizon from a height h, expanded for small h, and the error in a simpler model.
- Q10 (10 marks): an expansion used as a curve, then a proof that it never meets a second curve.
Key skills tested
The expansion. (1 + x) to the power n is 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ... for any rational n, valid when x is between −1 and 1.
Replacing x. For a power of (1 − 4x), use (−4x) in every term, with brackets, so (−4x)² = 16x² and (−4x)³ = −64x³.
Taking out a power. For (9 + 4x) to the power −½, take out 9 to the power −½, which is 1/3, not 9 or 3. Then expand (1 + 4x/9) to the power −½.
Validity. An expansion of a power of (a + bx) is valid when |x| is less than |a/b|. Use a strict inequality, and for two series take the narrower interval.
Unknown constants. Write each coefficient in terms of the unknowns and compare, such as nk = −2 and n(n−1)k²/2 = 6.
Products. For (a + bx) times an expansion, collect each power of x carefully, and keep only the terms asked for.
Partial fractions first. Split, then expand each part. For example, 3/(x − 2) = −3/2 times (1 − x/2) to the power −1.
Approximations. Choose x inside the valid interval, work out what the original expression equals there, which may be a reciprocal, then substitute into the series.
Accuracy. Percentage error is the size of (approximation − exact) ÷ exact × 100. The smaller x is, the better the approximation.
Worked example
Question 1 from this chapter. (a) Find the first four terms, in ascending powers of x, of the expansion of 1/√(9 + 4x), with each coefficient as a simplified fraction. (b) State where the expansion is valid.
(a) Write 1/√(9 + 4x) as (9 + 4x) to the power −½, and take out 9 to the power −½, which is 1/3. That leaves 1/3 times (1 + 4x/9) to the power −½.
Expanding with n = −½ and 4x/9 in place of x gives 1 − 2x/9 + 2x²/27 − 20x³/729. Multiplying every term by 1/3: 1/3 − 2x/27 + 2x²/81 − 20x³/2187.
(b) It is valid when |4x/9| < 1, which is |x| < 9/4.
Taking out 9 or 3 instead of 1/3 was the error the mark scheme singles out: the 1/3 must multiply every term.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2024, Q1: a binomial expansion with a negative x term inside the bracket, as in Q2 and Q8.
Where marks are lost
- Losing the minus sign. In January 2024 most errors came from using x or 4x in place of −4x in the expansion.
- Dropping the constant term. Occasionally the leading 1 was left out of the expansion.
- Expanding the wrong expression. A few rewrote the fraction and expanded the denominator instead, scoring no marks.
- Taking out the wrong factor. Taking out a instead of a to the power n changes every coefficient.
Common questions
Why does a fractional power need a range of validity?
Because the expansion never ends, and it only settles to the right value when the term inside the bracket is small.
Is the general binomial expansion in the formula booklet?
Yes, the expansion for (1 + x) to a power n is given, with its condition on x.
How do I pick x for an approximation?
Choose a value inside the valid range that makes the original expression equal the number you want, then substitute it.
Where this chapter leads
- Pure 4 Chapter 6: Integration: the partial fractions used here, integrated instead of expanded.
- Pure 4 Chapter 5: Differentiation: rates of change in models that also use approximations.