Binomial Expansion (AA HL)
Expanding a bracket like \((a+b)^n\) by hand for anything beyond squaring is impractical - the binomial theorem gives you the coefficients directly, using Pascal's triangle or \(\binom{n}{r}\). At HL you also meet an extended version that works for negative and fractional powers, which is where most of the exam marks in this sub-topic actually sit. It's part of the broader Binomial Theorem topic.
11 questions on this sub-topic.
The formulas
Covered under IB syllabus reference SL1.9. The binomial theorem gives a finite expansion for \((a+b)^n\) when \(n\) is a positive integer, using Pascal's triangle or the binomial coefficient \(\binom{n}{r}\), found either by formula or with your GDC.
Binomial theorem
\((a+b)^n=\displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)
Valid for any positive integer \(n\). The sum is finite - it has exactly \(n+1\) terms.
Binomial coefficient
\(\binom{n}{r}=\dfrac{n!}{r!(n-r)!}\)
Reads as "n choose r". Your GDC's \({}^nC_r\) function gives this instantly - no need to expand the factorials by hand.
Extended series
\((1+x)^n=1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots,\ |x|<1\)
For fractional or negative \(n\). The series never terminates, so it's only used to approximate - and only where \(|x|<1\).
Need the full syllabus wording and formula-booklet reference table? See Binomial Theorem.
Worked examples
Use the first three terms of the binomial expansion of \((1-2x)^{1/2}\) to find an approximate value of \(\sqrt{0.96}\), giving your answer to 3 significant figures.
Worked solution
\((1+u)^{1/2} = 1 + \tfrac12 u - \tfrac18 u^2 + \cdots\) with \(u = -2x\). M1
\((1-2x)^{1/2} = 1 - x - \tfrac12 x^2 - \cdots\) A1
\(x = 0.02.\) M1
\(1 - 0.02 - 0.0002 = 0.9798 \approx 0.980.\) A1
Find the first three terms in the expansion of \((4 - x)^{-2}\) in ascending powers of \(x\), and state the range of validity.
Worked solution
\((4-x)^{-2} = 4^{-2}\left(1 - \tfrac{x}{4}\right)^{-2}\) M1 \(= \tfrac{1}{16}\left(1 - \tfrac{x}{4}\right)^{-2}.\) A1
\((1+u)^{-2} = 1 - 2u + 3u^2 - \cdots\) with \(u = -\tfrac{x}{4}\): \(1 + \tfrac{x}{2} + \tfrac{3x^2}{16} + \cdots\) M1 A1
\((4-x)^{-2} = \tfrac{1}{16} + \tfrac{x}{32} + \tfrac{3x^2}{256} + \cdots\) A1 Valid for \(\left|\tfrac x4\right| < 1\), i.e. \(|x| < 4.\) A1
Common mistakes
- Ignoring the validity range \(|x|<1\). An infinite binomial expansion only means something where it converges - always state the range, especially when a question asks you to justify an approximation.
- Treating every expansion as infinite. The plain binomial theorem for a positive integer \(n\) gives a finite, exact expansion. Only fractional or negative \(n\) needs the extended series, and that result is an approximation, not an exact value.
- Dropping a sign when substituting \(u\). If the bracket contains a subtraction, like \((4-x)^{-2}\), you must substitute a negative \(u\) (here \(u=-\tfrac{x}{4}\)) into the series - forgetting the sign flips every odd-power term in the answer.
Ready to practise properly?
11 binomial-expansion questions, marked instantly like the real exam.
Quick answers
What is the binomial theorem?
For a positive integer \(n\), \((a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\), where \(\binom{n}{r}=\dfrac{n!}{r!(n-r)!}\) is the binomial coefficient, found by formula or with a GDC.
What is the extended binomial series used for?
It expands \((1+x)^n\) for fractional or negative \(n\), producing an infinite series that is only valid for \(|x|<1\) - unlike the finite expansion for a positive integer \(n\).