Binomial Theorem (AA SL)
Expanding \((a+b)^n\) by multiplying out brackets one at a time gets unmanageable fast once \(n\) is bigger than 3 or 4. The binomial theorem gives a direct formula for every term in the expansion, built from Pascal's triangle and the binomial coefficient \(nC_r\). This topic covers expanding a full binomial expression, and the more common exam skill of pulling out a single term or coefficient without expanding everything.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL1.9 | The binomial theorem: expansion of \((a+b)^n\), \(n\in\mathbb{N}\). Use of Pascal's triangle and \(nC_r\), found using both the formula and technology. |
This is a core AA syllabus point, examined on both papers.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the binomial theorem?
The binomial theorem is a formula for expanding \((a+b)^n\) directly into a sum of terms, without multiplying out brackets one at a time. Each term has the form \(nC_r\, a^{n-r}b^r\), with \(r\) running from \(0\) to \(n\).
e.g. \((a+b)^2 = a^2 + 2ab + b^2\), which matches \(nC_0a^2+nC_1ab+nC_2b^2\) with \(n=2\).
What is a binomial coefficient?
A binomial coefficient \(nC_r\), also written \(\binom{n}{r}\), counts the number of ways to choose \(r\) items from a set of \(n\), and it's exactly the coefficient that appears in front of each term of a binomial expansion.
e.g. \(\binom{4}{2} = \dfrac{4!}{2!\,2!} = 6\).
What is Pascal's triangle?
Pascal's triangle is a triangular array of numbers where each entry is the sum of the two above it, and row \(n\) gives the binomial coefficients \(nC_0, nC_1, \ldots, nC_n\) needed for expanding \((a+b)^n\).
e.g. Row 4 of Pascal's triangle is \(1,4,6,4,1\), matching the coefficients in \((a+b)^4\).
What is the general term?
The general term \(T_{r+1} = nC_r\, a^{n-r}b^r\) represents the \((r+1)\)th term in the expansion of \((a+b)^n\). It's the key tool for finding one specific term - such as the term in \(x^5\) - without expanding the whole expression.
e.g. In \((2x+3)^7\), the term in \(x^5\) needs \(7-r=5\), so \(r=2\), giving \(T_3=\binom{7}{2}(2x)^5(3)^2\).
What is a constant term?
The constant term of an expansion is the term where every power of the variable \(x\) has cancelled out, so it's just a number. It usually comes from a bracket like \(\left(x+\tfrac{k}{x}\right)^n\), where the exponents of \(x\) can add to zero.
e.g. In \(\left(x+\tfrac{2}{x}\right)^6\), the constant term needs \(6-2r=0\), so \(r=3\), giving \(\binom{6}{3}2^3=160\).
Key formulas
Two formulas cover this entire topic. The tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
Both the binomial expansion and the \(nC_r\) formula are given in the official formula booklet.
| Formula | Used for | Booklet? |
|---|---|---|
| \((a+b)^n=\displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\) | Full binomial expansion | ✓ Yes |
| \(\displaystyle\binom{n}{r}=\dfrac{n!}{r!\,(n-r)!}\) | Binomial coefficient \(nC_r\) | ✓ Yes |
| \(T_{r+1}=\binom{n}{r}a^{n-r}b^r\) | General term (for a specific term) | Follows from the booklet formula |
Full expansion vs specific term
Most exam questions don't want the whole expansion - they want one term. This table sets out the difference in approach.
| Feature | Full expansion | Specific term |
|---|---|---|
| What you find | Every term, \(r=0\) to \(n\) | One value of \(r\) only |
| Method | Substitute each \(r\) in turn | Solve for the \(r\) that matches the target power |
| Typical prompt | "Expand \((x+2)^4\)" | "Find the coefficient of \(x^5\)" |
| Speed | Slower - every term needed | Faster - only the general term is needed |
Applying the theorem
These are the three situations the theorem is used for on the AA syllabus.
Full expansion
\[(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\]
Substitute \(r=0,1,\ldots,n\) in turn, using Pascal's triangle for the coefficients.
✓ In the formula bookletA specific term
\[T_{r+1}=\binom{n}{r}a^{n-r}b^r\]
Set the power of the target variable equal to the power you want, solve for \(r\), then substitute.
Follows directly from the booklet formulaAn unknown coefficient
\[\binom{n}{r}a^{n-r}b^r = \text{given value}\]
Set the coefficient expression equal to the value given in the question, then solve the resulting equation for the unknown.
Combines the booklet formula with equation-solvingWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Use the binomial theorem to expand \((x+2)^4\), giving your answer in descending powers of \(x\).
Worked solution
\((x+2)^4=\binom40x^4+\binom41x^3(2)+\binom42x^2(4)+\binom43x(8)+\binom44(16)\) M1
\(=x^4+8x^3+24x^2+32x+16.\) A1
\(x^4+8x^3+24x^2+32x+16.\) A1
In the expansion of \((1+ax)^8\), the coefficient of \(x^2\) is \(112\).
(a) Show that \(28a^2=112\).
(b)(i) Find the value of \(a<0\).
(b)(ii) Find the value of \(a>0\).
(c) For the positive value of \(a\), find the coefficient of \(x^3\).
Worked solution
(a) Coefficient of \(x^2\) is \(\binom82 a^2=28a^2.\) M1 A1
So \(28a^2=112.\) AG
(b) \(a^2=4\Rightarrow a=-2\) A1 or \(a=2.\) A1
(c) \(a=2:\) coefficient of \(x^3=\binom83 a^3=56\cdot8=448.\) M1 A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Forgetting to raise the whole term to the power, not just the variable. In \((2x+3)^7\), the term needs \((2x)^{7-r}\), not \(2\cdot x^{7-r}\) - the coefficient 2 gets raised to the power too.
- Miscounting which term corresponds to which value of \(r\). The general term \(T_{r+1}\) has index \(r+1\), so the "3rd term" is \(r=2\), not \(r=3\) - always work from the power of \(x\) you want, not the position in the sequence.
- Losing a sign when the second term is negative. In \((2x-1)^5\), the term is \((2x)^{5-r}(-1)^r\) - dropping the \((-1)^r\) factor silently turns every other term positive when it should be negative.
- Solving for \(r\) but forgetting to substitute back. Finding \(r=3\) answers "which term?", not "what is the coefficient?" - the value of \(r\) still needs to go back into \(\binom{n}{r}a^{n-r}b^r\) to get the actual numerical answer.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Count arrangements and selections quickly - and get the exact binomial coefficients used in the binomial theorem, without building out Pascal's triangle row by row.
- Type n, then MATH → PROB → nCr (or nPr), then r, then ENTER.TI-84
- Use nCr(n, r) or nPr(n, r) from the catalog.Nspire
- Type n, then OPTN → PROB → nCr (or nPr), then r.Casio
Tip: nCr(n, r) is exactly the binomial coefficient \(\binom{n}{r}\), so this is the fastest way to check a coefficient once you've found the correct value of \(r\) algebraically.
For very large or very small numbers - avoids typing long strings of zeros and prevents rounding errors, which matters once a coefficient from a high power of \(n\) gets large.
- Scientific notation means \(a\times10^n\), e.g. \(3.2\times10^8\) or \(4.5\times10^{-3}\).
- Use 2nd → , (EE) to enter the ×10 part: type 3.2 2nd , 8 to enter 3.2×10⁸. Do NOT type ×10^ separately.TI-84
- Use the EE key (or type ×10^ from the keyboard template) to enter scientific notation. Or just type 3.2×10^8 using the ^ key.Nspire
- Use the ×10ᴉ key (EXP key) - type 3.2 then EXP then 8. Do NOT type ×10^ manually.Casio
- To display answers in scientific notation: on TI-84 press MODE and choose SCI; on Casio set the display mode in SET UP.
Tip: A common mistake is typing ×10^ instead of using the EE/EXP key - this gives ×10×... (multiplication, then a power) rather than proper scientific notation.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Binomial theorem questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What does nCr actually mean?
nCr, written \(\binom{n}{r}\), counts the number of ways to choose \(r\) items from \(n\) without regard to order. In the binomial expansion it gives the coefficient of the term where the second bracket's power is \(r\).
How do I find one specific term without expanding everything?
Use the general term \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\). Set the power of the variable you want equal to the target power, solve for \(r\), then substitute that \(r\) back in to get the coefficient.
Is the binomial theorem in the formula booklet?
Yes - both the expansion of \((a+b)^n\) and the formula for \(nC_r\) are given in the booklet, so you don't need to memorise them, but you do need to know how to apply them to find a specific term.
Can I use my GDC for this topic?
Yes - your calculator has a built-in nCr function that computes binomial coefficients instantly, which is much faster and safer than working them out from Pascal's triangle by hand for larger values of \(n\). See the GDC guide for model-specific instructions.
Sub-topics
Binomial Theorem broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA SL syllabus unit, in case you want to keep going.