Cambridge A Level Mathematics 9709 · Pure 1

Binomial expansion

The binomial expansion writes out powers of a two-term bracket without multiplying it repeatedly. For a positive integer $n$ — the Pure 1 case —

(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r}

Most P1 binomial questions don’t want the whole expansion: they want one specific term or coefficient, and Cambridge’s examiner reports repeatedly note candidates expanding everything (wasting time) or forgetting to raise the $b$ -part’s coefficient to its power (losing the marks).

Updated 20 August 2026

The coefficients: nCr

The numbers $\binom{n}{r}$ (said “n choose r”, calculator key nCr) are the rows of Pascal’s triangle:

\binom{n}{r} = \frac{n!}{r!\,(n-r)!}, \qquad \binom{6}{2} = 15

For small $n$ , Pascal’s triangle by hand is fine; for $(2-\tfrac{x}{2})^{10}$ -style questions, nCr is faster and less error-prone. Symmetry helps too: $\binom{n}{r} = \binom{n}{n-r}$ .

The general term — the tool that answers everything

The term containing $b^r$ in $(a+b)^n$ is

T_{r+1} = \binom{n}{r} a^{n-r} b^{r}

Every “find the coefficient of $x^k$ ” question is this formula plus one line of index work: set the power of $x$ inside $b^r$ equal to $k$ , solve for $r$ , substitute. The critical discipline: $a$ and $b$ are the whole terms, coefficients and signs included. In $(2-3x)^5$ , $a = 2$ and $b = -3x$ — so $b^r$ carries $(-3)^r$ , not just $x^r$ .

Worked example

Find the coefficient of x³ in the expansion of (2 − 3x)⁵.

Step 1 — set up the general term with $a = 2$ , $b = -3x$ , $n = 5$ :

T_{r+1} = \binom{5}{r}\, 2^{5-r} (-3x)^r

Step 2 — pick out x³: need $r = 3$ .

\binom{5}{3}\, 2^{2} (-3)^3 x^3 = 10 \times 4 \times (-27)\, x^3 = -1080x^3

Answer: the coefficient is $-1080$ . Note all three factors were needed — the nCr, the power of 2, and the cube of $-3$ with its sign. Each is typically worth working credit; omitting $(-3)^3$ and writing $-3$ is the classic slip.

Finding unknowns from a given coefficient

The harder P1 variant runs the machine backwards: “the coefficient of $x^2$ in $(1+ax)^6$ is 60; find $a$ .” Same general term:

\binom{6}{2} a^2 = 15a^2 = 60 \;\Rightarrow\; a^2 = 4 \;\Rightarrow\; a = \pm 2

Keep both roots unless the question restricts $a$ (“where $a > 0$ ”). Examiner reports specifically call out dropped negative roots here. A second common variant multiplies two expansions — e.g. $(1+2x)(1-x)^7$ — where the target power collects from two products: constant × $x^k$ term plus $x$ term × $x^{k-1}$ term.

The mistakes that lose the marks

1.Forgetting to raise b's coefficient to the power
In $(2-3x)^5$ , the $x^3$ term carries $(-3)^3 = -27$ , never $-3$ . Bracket the whole term — $(-3x)^r$ — before separating number from letter.
2.Losing the sign in (a − b)ⁿ
Treat the minus as part of $b$ : $b = -3x$ . Odd powers of $b$ are negative, even powers positive — if your expansion doesn’t alternate, something’s wrong.
3.Expanding the whole bracket when one term was asked
“Find the coefficient of $x^3$ ” needs the general term and one value of $r$ — three lines. A full ten-term expansion invites arithmetic slips and burns minutes the paper doesn’t give back.
4.Confusing 'coefficient' with 'term'
The term is $-1080x^3$ ; the coefficient is $-1080$ . Answer the noun the question used.
5.Dropping the ± when solving for an unknown
$a^2 = 4$ has two solutions. Write $a = \pm 2$ , then discard one only if the question gives a reason to.

Common questions

Is the binomial expansion for negative or fractional powers in Pure 1?

No — P1 covers positive integer $n$ only, where the expansion is exact and finite. The infinite series for negative and fractional powers (with its validity condition $|x| < 1$ ) belongs to Pure 3.

Do I have to simplify every coefficient?

Yes — mark schemes state coefficients as evaluated numbers ( $-1080$ , not $10 \times 4 \times (-27)$ ). Leave the arithmetic line visible, then finish it.

How is the binomial expansion examined in 9709 P1?

Usually one 4–6 mark question: a specific coefficient, an unknown constant from a given coefficient, or a two-bracket product where the target power collects from two places. It also feeds series questions — the first terms of an expansion reappearing as a geometric or arithmetic progression.

Practise binomial expansion against real mark schemes

Quanta has real Cambridge A Level Maths 9709 past-paper questions, broken into skill checkpoints, marked criterion by criterion the way examiners mark — and it tracks which skills you’re missing. Free for individual students.

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