In the expansion of $\left(\frac{1}{2}x + 1\right)^n$, the coefficient of the $x^2$ term is $8n$, where $n \in \mathbb{Z}^+$. Find the exact value of $n$.

1. Write the formula for the $x^2$ term using the general expansion: $\binom{n}{2}\left(\frac{1}{2}x\right)^2(1)^{n-2}$. 2. Expand the binomial coefficient algebraically: $\binom{n}{2} = \frac{n(n-1)}{2!}$. 3. Simplify the $x^2$ term expression: $\frac{n(n-1)}{2} \times \frac{1}{4}x^2 = \frac{n^2 - n}{8}x^2$. 4. Equate this coefficient to the given value of $8n$: $\frac{n^2 - n}{8} = 8n$. 5. Rearrange into a quadratic equation: $n^2 - n = 64n \implies n^2 - 65n = 0 \implies n(n - 65) = 0$. 6. Solve for $n$. Since $n$ must be a positive integer, $n \neq 0$, so $\mathbf{n = 65}$.

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IB Maths AA SL · Unit 1: Number and Algebra

Binomial Theorem (IB Maths AA SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

The binomial theorem expands $(a + b)^{n}$ for a positive whole number $n$ , with coefficients $(\binom{n}{r})$ from Pascal's triangle or your calculator. Most AA SL questions want one term: write the general term $(\binom{n}{r}) a^{n - r} b^{r}$ , choose $r$ to get the power you need, and raise every number inside the bracket to its power.

46 questions
Paper 1: 39
Paper 2: 7
3 worked examples

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Binomial Theorem notes

Expanding $(a + b)^{n}$

$(a + b)^{n} = a^{n} + (\binom{n}{1}) a^{n - 1} b + (\binom{n}{2}) a^{n - 2} b^{2} + \dots + b^{n}$ . The powers of $a$ fall by one each term while the powers of $b$ rise, and every term has total power $n$ . For example $(x + 2)^{3} = x^{3} + 6 x^{2} + 12 x + 8$ .

$(\binom{n}{r}) = \frac{n!}{r! (n - r)!}$ is row $n$ of Pascal's triangle; on Paper 2 use the nCr key.

The general term

The term with $b^{r}$ is $(\binom{n}{r}) a^{n - r} b^{r}$ . It is not printed in the booklet, so learn it. Put the whole of each part in brackets: in $(2 x - 1)^{5}$ , $a = 2 x$ and $b = - 1$ .

Finding a specific coefficient

For the coefficient of $x^{3}$ in $(2 x - 1)^{5}$ you need $(2 x)^{3}$ , so $r = 2$ : $(\binom{5}{2}) (2 x)^{3} (- 1)^{2} = 10 \times 8 x^{3} = 80 x^{3}$ . The coefficient is $80$ . Forgetting to cube the 2 is the most common lost mark.

The constant term

When the bracket mixes $x$ and $\frac{1}{x}$ , find the term where the powers cancel. In ${(x + \frac{2}{x})}^{4}$ the general term has $x^{4 - r} \cdot x^{- r} = x^{4 - 2 r}$ , so $r = 2$ and the constant term is $(\binom{4}{2}) \times 2^{2} = 24$ .

Finding n or an unknown

If a coefficient is given, write the general term with the unknown and solve. If the coefficient of $x^{2}$ in $(1 + x)^{n}$ is $28$ , then $\frac{n (n - 1)}{2} = 28$ , so $n = 8$ (reject the negative root).

Key formulas

Binomial theorem, $n \in N$ In the booklet: $(a + b)^{n} = a^{n} + (\binom{n}{1}) a^{n - 1} b + \dots + (\binom{n}{r}) a^{n - r} b^{r} + \dots + b^{n}$
Binomial coefficient In the booklet: $(\binom{n}{r}) =^{n} C_{r} = \frac{n!}{r! (n - r)!}$
General term Learn it: $T_{r + 1} = (\binom{n}{r}) a^{n - r} b^{r}$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Explained, with a worked example: Binomial expansion

Binomial Theorem worked examples

Worked example 1: Finding the first few terms · easy

Find the first three terms, in ascending powers of $x$ , in the binomial expansion of $(3 + x)^{4}$ .

Solution

5 steps

Examiner tip: A common mistake is forgetting to apply the combination coefficient

(\binom{n}{r})

entirely, simply writing the terms as

a^{n} + a^{n - 1} b + a^{n - 2} b^{2}

. Always check against Pascal's triangle!

Worked example 2: Finding a specific coefficient · medium

The coefficient of $x^{7}$ in the expansion of $x^{3} (a x + 3)^{5}$ is $1215$ . Find the possible values of the real constant $a$ .

Solution

6 steps · 1 to fill in

Examiner tip: When dealing with terms like

(a x)^{4}

, students frequently forget to apply the power to the constant

a

, writing

a x^{4}

instead of

a^{4} x^{4}

Worked example 3: Solving for the power $n$ · hard

In the expansion of ${(\frac{1}{2} x + 1)}^{n}$ , the coefficient of the $x^{2}$ term is $8 n$ , where $n \in Z^{+}$ . Find the exact value of $n$ .

Solution

6 steps · 1 to fill in

Examiner tip: Many students struggle with expanding

(\binom{n}{2})

algebraically. Remember it is simply

\frac{n (n - 1)}{2}

. Do not divide by

n

to solve the quadratic; always factorise to avoid losing roots, even if

n = 0

is later rejected.

Binomial Theorem exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 3 marks · Paper 1

Find the first three terms, in ascending powers of $x$ , in the expansion of $(9 - 2 x)^{5}$ .

Check your answer in practice →

Question 2 · medium · 5 marks · Paper 1

In the expansion of $(3 + p x)^{6}$ , the coefficient of the $x^{4}$ term is four times the coefficient of the $x^{2}$ term. Find the possible values of $p$ .

Full worked solution Free, no sign-in

Write down the $x^{2}$ term: $(\binom{6}{2}) 3^{4} (p x)^{2} = 15 (81) p^{2} x^{2} = 1215 p^{2} x^{2}$ . (A1)
Write down the $x^{4}$ term: $(\binom{6}{4}) 3^{2} (p x)^{4} = 15 (9) p^{4} x^{4} = 135 p^{4} x^{4}$ . (A1)
Set the $x^{4}$ coefficient equal to four times the $x^{2}$ coefficient: $135 p^{4} = 4 (1215 p^{2})$ . (M1)
Rearrange and factorise: $135 p^{4} = 4860 p^{2} ⟹ p^{4} - 36 p^{2} = 0 ⟹ p^{2} (p^{2} - 36) = 0$ . (M1)
$p = 0$ makes both coefficients zero (trivial case), so $p = \pm 6$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

Consider the expansion of ${(2 x + \frac{k}{x})}^{9}$ , where $k > 0$ . The coefficient of the term in $x^{3}$ is equal to the coefficient of the term in $x^{5}$ . Find the exact value of $k$ .

Attempt it and see the mark scheme →

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Where students lose marks on binomial theorem

Examiner's noteBinomial expansion: the coefficient inside the bracket is not raised to the power

Where marks are lost: Reported in two sessions: (2x)⁶ written as 2x⁶, brackets omitted round the 2x, the whole term given when only the coefficient was asked, and a second valid value of k missed.

Examiners flagged this in 2 exam series (November 2024, May 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Binomial Theorem FAQ

Is the binomial theorem in the IB formula booklet?

Yes: the expansion of (a + b)^n and the formula for nCr are in the AA SL booklet. The general term nCr a^(n-r) b^r is not printed, so learn it: most questions are about one term.

Do I need Pascal's triangle?

It is quick for small n (up to about 6). For larger n, use nCr from the formula, or the nCr key on your calculator on Paper 2.

How many IB Maths AA SL binomial theorem questions are there?

There are 46 exam-style binomial theorem questions in the AA SL question bank (Paper 1: 39 · Paper 2: 7), graded 10 easy, 19 medium, 12 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is binomial theorem on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 39 · Paper 2: 7. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 1: Exponentials & Logarithms · Financial Math · Proof · Sequences & Series

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Binomial expansion (positive integer n) · Specific coefficients & terms.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL number and algebra knowledge organiser

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Binomial Theorem (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Binomial Theorem notes

Expanding (a+b)n

The general term

Finding a specific coefficient

The constant term

Finding n or an unknown

Key formulas

Binomial Theorem worked examples

Worked example 1: Finding the first few terms · easy

Solution

Worked example 2: Finding a specific coefficient · medium

Solution

Worked example 3: Solving for the power n · hard

Solution

Binomial Theorem exam-style questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Where students lose marks on binomial theorem

Binomial Theorem FAQ

Next steps

What else is free here

Expanding $(a + b)^{n}$

Worked example 3: Solving for the power $n$ · hard