Binomial Expansion Explained: A-Level Maths (9709)

Short answer: Binomial expansion is a fast way to expand a bracket like $(a+b)^n$ without multiplying it out term by term. For a positive whole-number power you use $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$, which gives a finite expansion. For negative or fractional powers (Pure 3), you use the extended version, which gives an infinite series that is only valid when $|x|$ is small enough.

The binomial theorem for positive integer powers

In Cambridge 9709 Pure Mathematics 1, you expand $(a+b)^n$ where $n$ is a positive whole number. The theorem is:

$(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n}b^n$

The symbol $\binom{n}{r}$, read as "n choose r", is calculated as $\binom{n}{r} = \frac{n!}{r!\,(n-r)!}$. Your calculator has this as the nCr button, which saves a lot of time in the exam.

Notice the pattern: the power of $a$ goes down by one each term while the power of $b$ goes up by one, and the two powers always add up to $n$.

Finding a specific term

Most 9709 questions do not ask for the whole expansion. They ask for one term, such as "the coefficient of $x^3$". Expanding everything wastes time, so use the general term:

$\text{term in } b^r = \binom{n}{r} a^{n-r} b^r$

Worked example. Find the coefficient of $x^3$ in $(1 + 2x)^{10}$.

Here $a = 1$, $b = 2x$, and $n = 10$. The term in $x^3$ comes from $r = 3$:

$\binom{10}{3}(1)^{7}(2x)^{3} = 120 \times 8x^3 = 960x^3$

So the coefficient of $x^3$ is $960$. The key steps are choosing the right $r$, remembering to raise the whole of $2x$ to the power, and then evaluating $2^3 = 8$. Forgetting that last cube is one of the most common slips.

When there is more than one source of the power

If both terms in the bracket contain $x$, say $(3x - \frac{2}{x})^n$, then the power of $x$ in a term depends on $r$. You set up the general term, simplify the powers of $x$, set the total power equal to the one you want, and solve for $r$. Only then do you evaluate. This is a favourite exam trick because it forces you to track the powers carefully rather than just reading off a coefficient.

The extended binomial expansion (Pure 3)

In Pure Mathematics 3 the power can be negative or a fraction. The nCr formula no longer works because $n$ is not a whole number, so you use:

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$

This series never terminates. It only equals $(1+x)^n$ when $|x| < 1$.

Worked example. Expand $(1+x)^{1/2}$ up to the term in $x^2$.

With $n = \tfrac{1}{2}$:

• First term: $1$
• Second term: $\tfrac{1}{2}x$
• Third term: $\dfrac{\tfrac{1}{2}\left(\tfrac{1}{2}-1\right)}{2!}x^2 = \dfrac{\tfrac{1}{2} \times \left(-\tfrac{1}{2}\right)}{2}x^2 = -\tfrac{1}{8}x^2$

So $(1+x)^{1/2} \approx 1 + \tfrac{1}{2}x - \tfrac{1}{8}x^2$, valid for $|x| < 1$.

Getting the bracket into the right form

The extended formula only works when the bracket starts with a $1$. If you are given $(4 + x)^{1/2}$, first factor out the constant:

$(4+x)^{1/2} = \left(4\left(1 + \tfrac{x}{4}\right)\right)^{1/2} = 2\left(1 + \tfrac{x}{4}\right)^{1/2}$

Now expand $(1 + \tfrac{x}{4})^{1/2}$ using the formula and multiply every term by $2$. The validity condition also changes: you need $\left|\tfrac{x}{4}\right| < 1$, which means $|x| < 4$.

Comparing the two versions

	Positive integer power (P1)	Extended power (P3)
Value of $n$	Whole number	Negative or fractional
Number of terms	Finite ($n+1$ terms)	Infinite series
Uses $\binom{n}{r}$	Yes	No, uses the fraction formula
Validity condition	Always valid	Only when $\lvert x\rvert < 1$
Bracket must start with 1	No	Yes, factor out first

Common mistakes to avoid

• Forgetting to raise the coefficient inside the bracket to the power, for example writing $2x^3$ instead of $(2x)^3 = 8x^3$.
• Losing a minus sign when the bracket has a subtraction, such as $(1 - 3x)^n$.
• Quoting the validity condition for the factored bracket but forgetting to translate it back into a condition on $x$.
• Reading "coefficient" as "term" or the other way round. A coefficient is just the number, so $960$, not $960x^3$.

Practise until the steps are automatic

Binomial questions reward accuracy more than cleverness, so the fastest way to improve is to do many of them and check each line. ExamPal is an AI tutor that already knows the 9709 syllabus and marks your working against the Cambridge mark scheme, so if you drop a coefficient or misstate a validity range it can show you exactly where and why. That kind of targeted correction is how the small slips stop happening.