Binomial Theorem (AA HL)
The binomial theorem gives a shortcut for expanding \((a+b)^n\) without multiplying out every bracket by hand. This topic covers the theorem for positive integer powers using Pascal's triangle and \(\binom{n}{r}\), and its HL extension to fractional and negative indices - an infinite series that's only valid for restricted values of \(x\), and is often used to approximate roots and reciprocals.
What the syllabus says
This topic maps onto two points 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 \(\binom{n}{r}\), found using both the formula and technology. |
| AHL1.10 | Counting principles, including permutations and combinations. Extension of the binomial theorem to fractional and negative indices, i.e. \((a+b)^n,\ n\in\mathbb Q\), using \((a+b)^n=a^n(1+\tfrac{b}{a})^n\). |
SL1.9 is also examinable at AA HL; AHL1.10 extends it to indices that aren't positive integers.
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 gives a formula for expanding \((a+b)^n\) as a sum of terms \(\binom{n}{r}a^{n-r}b^r\), without multiplying out \(n\) brackets one at a time. For small \(n\) the coefficients are just the corresponding row of Pascal's triangle.
e.g. \((1+x)^3=1+3x+3x^2+x^3\), using coefficients \(1,3,3,1\).
What is a binomial coefficient \(\binom{n}{r}\)?
\(\binom{n}{r}\) (read "\(n\) choose \(r\)") counts the number of ways to choose \(r\) objects from \(n\), and equals \(\dfrac{n!}{r!(n-r)!}\). In an expansion it tells you how many of the \(n\) brackets contribute the \(b\) rather than the \(a\).
e.g. \(\binom{5}{2}=\dfrac{5!}{2!\,3!}=10\).
What is Pascal's triangle?
Pascal's triangle is a triangular array where each entry is the sum of the two above it, and row \(n\) gives every binomial coefficient \(\binom{n}{r}\) for that power. It's a fast way to find coefficients without computing factorials.
e.g. Row \(n=4\) reads \(1,4,6,4,1\), matching \(\binom{4}{0},\binom{4}{1},\ldots,\binom{4}{4}\).
What does "extension to fractional and negative indices" mean?
When \(n\) isn't a positive integer, \((1+x)^n\) can't be written as a finite product of brackets - instead it becomes an infinite series, valid only for \(|x|<1\), built from the same coefficient pattern generalised to non-integer \(n\).
e.g. \((1+x)^{-1}=1-x+x^2-x^3+\cdots\) for \(|x|<1\).
How do you use a binomial expansion to approximate a value?
Substitute a small value of \(x\) into a truncated expansion, so the higher powers of \(x\) become negligible and the first few terms give a close approximation to a root or reciprocal.
e.g. \((1+x)^{1/2}\approx1+\tfrac12x\), so with \(x=0.02\): \(\sqrt{1.02}\approx1+0.01=1.01\) (true value \(\approx1.0100\)).
Key formulas
Four formulas cover almost every question on this topic. The two tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The general term, the \(\binom{n}{r}\) formula and the extended series are all printed in the official formula booklet.
| Formula | Used for | Booklet? |
|---|---|---|
| \((a+b)^n=\displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\) | Binomial theorem, positive integer \(n\) | ✓ Yes |
| \(\binom{n}{r}=\dfrac{n!}{r!(n-r)!}\) | Binomial coefficient | ✓ Yes |
| \((1+x)^n=1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots,\ |x|<1\) | Extended binomial theorem, fractional/negative \(n\) | ✓ Yes |
| \(\displaystyle\sum_{r=0}^{n}\binom{n}{r}=2^n\) | Sum of a row of Pascal's triangle | Not in booklet - derived by substituting \(x=1\) |
Integer vs extended binomial theorem
The two versions look similar but behave very differently once \(n\) stops being a positive integer.
| Feature | Positive integer \(n\) | Fractional/negative \(n\) |
|---|---|---|
| Expansion | Finite - exactly \(n+1\) terms | Infinite series |
| Validity | True for every value of \(x\) | Only valid for \(|x|<1\) (or a scaled bound) |
| Coefficients | \(\binom{n}{r}\), computable with the nCr button | \(\dfrac{n(n-1)\cdots(n-r+1)}{r!}\) - no nCr button, since \(n\) isn't an integer |
The binomial theorem for positive integers
The full theorem describes every term in the expansion, but most questions only need one or two of them.
General term
\[T_{r+1}=\binom{n}{r}a^{n-r}b^r\]
The \((r+1)\)th term - use it to jump straight to a specific term without expanding the whole bracket.
✓ In the formula bookletPascal's triangle
Each row starts and ends in 1; every other entry is the sum of the two numbers above it in the previous row. Quick for small \(n\), impractical for large \(n\).
Finding a specific coefficient
Set the powers of \(a\) and \(b\) in the general term equal to what the question asks for, solve for \(r\), then substitute back to get the coefficient.
Extending to fractional and negative indices
The same coefficient pattern generalises once \(n\) is no longer restricted to non-negative integers - but the series no longer terminates.
The extended series
\[(1+x)^n=1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots\]
Works for any real \(n\); when \(n\) is a positive integer it naturally terminates and matches the standard theorem.
✓ In the formula bookletValidity condition
The series only converges for \(|x|<1\) - always state this range, as it's typically worth a mark on its own (an R1).
Rewriting \((a+b)^n\)
\[(a+b)^n=a^n\left(1+\dfrac{b}{a}\right)^n\]
Needed whenever the bracket isn't already in the form \((1+x)^n\) - factor out \(a^n\) first, then expand the remaining \((1+b/a)^n\).
Using expansions to approximate
A binomial expansion truncated after two or three terms is often accurate enough for an approximation, provided \(x\) is small.
Choosing the substitution
Match the target expression to \((1+x)^n\) form and solve for the value of \(x\) that makes the bracket equal your target number.
Truncating the series
Keep only as many terms as the question specifies (e.g. "up to the term in \(x^2\)"); the rest are assumed negligible for small \(x\).
Checking accuracy
Compare the approximation to the exact value on your GDC - a good approximation should agree to several significant figures.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Consider \((2x-1)^4.\)
(a) Expand it using the binomial theorem.
(b) Verify your expansion by evaluating both the original expression and the expansion at \(x=1.\)
Worked solution
(a) \((2x-1)^4=16x^4-32x^3+24x^2-8x+1.\) M1
A1
A1
(b) \((2(1)-1)^4=1^4=1;\) expansion gives \(16-32+24-8+1=1.\) Both agree. R1
Consider the binomial expansion of \((1 - 3x)^{1/3}.\)
(a) Find the expansion up to and including the term in \(x^2\).
(b) By substituting a suitable value of \(x\), find an approximation to \(\sqrt[3]{0.97}.\)
Worked solution
(a) Expand to the \(x^2\) term.
\((1-3x)^{1/3} = 1 + \tfrac13(-3x) + \dfrac{\tfrac13(-\tfrac23)}{2}(-3x)^2 + \cdots\) M1
\(= 1 - x - x^2 - \cdots\) A1 A1
(b) Approximate \(\sqrt[3]{0.97}.\)
\(1 - 3x = 0.97 \Rightarrow x = 0.01.\) M1
\(\sqrt[3]{0.97} \approx 1 - 0.01 - (0.01)^2 = 0.9899.\) 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 the alternating signs in \((a-b)^n\). Writing \((a-b)^n\) as \((a+(-b))^n\) means every odd-power term picks up a minus sign - it's easy to drop one partway through.
- Using \(\binom{n}{r}\) for a fractional or negative index. The nCr formula only works for non-negative integer \(n\) - once \(n\) is fractional or negative, you must use the extended series coefficients instead.
- Ignoring the validity range \(|x|<1\). An infinite binomial expansion is only meaningful where it converges - always state the range, especially when asked to justify an approximation.
- Forgetting to factor out \(a^n\) first. If the bracket isn't already \((1+x)^n\), you must rewrite \((a+b)^n\) as \(a^n(1+b/a)^n\) before applying the extended series - skipping this step gives the wrong coefficients.
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 coefficients used in the binomial theorem.
- 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 ignores order (choosing a team); nPr counts order (ranking places). nCr(n, r) is the binomial coefficient.
For very large or very small numbers thrown up by expansions with big coefficients - avoids typing long strings of zeros and prevents rounding errors.
- 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 \(3.2\times10^8\). 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 ×10ˣ 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's the difference between the binomial theorem and the extended binomial theorem?
The standard binomial theorem expands \((a+b)^n\) for a positive integer \(n\), giving a finite sum of \(n+1\) terms that's exactly true for every value of \(x\). The extended version handles fractional or negative \(n\), giving an infinite series that's only a valid approximation when the expansion converges.
Why does the extended binomial expansion need \(|x|<1\)?
With a fractional or negative index, the series never terminates - it keeps generating more terms forever. The series only converges to a finite value when \(|x|\) is small enough, specifically less than 1 (or a scaled bound if the bracket isn't exactly \(1+x\)).
How do I find a specific term in an expansion without expanding everything?
Use the general term formula, \(T_{r+1}=\binom{n}{r}a^{n-r}b^r\). Substitute the value of \(r\) that gives the power of \(x\) you want, without writing out any of the surrounding terms.
Can I use my GDC to find binomial coefficients?
Yes - every GDC has an nCr button that computes the binomial coefficient directly, which is much faster than building 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 HL syllabus unit, in case you want to keep going.