Binomial Expansion (AA SL)
Expanding \((a+b)^n\) by hand looks slow until you spot the pattern: the coefficients follow Pascal's triangle, and the powers of \(a\) and \(b\) move in opposite directions across the expansion. This page focuses purely on producing a full expansion - with worked examples and the mistakes that lose the most marks. It's part of the broader Binomial Theorem topic.
14 questions on this sub-topic.
The formula
Covered under IB syllabus reference SL1.9: the binomial theorem for \((a+b)^n\), \(n\in\mathbb{N}\), using Pascal's triangle and \(nC_r\), found using both the formula and technology.
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 bookletBinomial coefficient
\(\displaystyle\binom{n}{r}=\dfrac{n!}{r!\,(n-r)!}\)
This is the same value \(nC_r\) that your GDC's combinations function gives you - useful for checking, but you should still be able to build a full expansion by hand.
Need the full syllabus wording and formula-booklet reference table? See Binomial Theorem.
Worked examples
Expand and simplify \((1-3x)^4\).
Worked solution
Row 4 of Pascal's triangle is \(1,4,6,4,1\); apply these to \((1+(-3x))^4\), noting that the sign alternates through the powers of \(-3x\). M1
\(1+4(-3x)+6(-3x)^2+4(-3x)^3+(-3x)^4.\) A1
\(=1-12x+54x^2-108x^3+81x^4.\) A1
Expand \((1+\sqrt2)^3\), giving your answer in the form \(a + b\sqrt2\).
Worked solution
Pascal's row for \(n=3\) is \(1,3,3,1\), so \((1+\sqrt2)^3=1+3\sqrt2+3(\sqrt2)^2+(\sqrt2)^3.\) M1
Since \((\sqrt2)^2=2\) and \((\sqrt2)^3=2\sqrt2\), this becomes \(1+3\sqrt2+6+2\sqrt2.\) A1A1
Collecting like terms, \(=7+5\sqrt2.\) A1
Find the coefficient of \(x^2\) in the expansion of \((1 + 2x)^5 (1 - x)^4.\)
Worked solution
\(1 + 10x + 40x^2 + \cdots\) (\(\binom52 2^2\) M1 \(= 40\)). A1
\(1 - 4x + 6x^2 + \cdots\) M1 A1
\((1)(6) + (10)(-4) + (40)(1)\) M1 \(= 6 - 40 + 40 = 6.\) A1
Expand and simplify \((2 + x)^4\) in ascending powers of \(x\).
Worked solution
\((2+x)^4 = \binom40 2^4 + \binom41 2^3 x + \binom42 2^2 x^2 + \binom43 2\,x^3 + \binom44 x^4.\) M1
\(16 + 32x + 24x^2 + 8x^3 + x^4.\) A1 A1 A1
Common mistakes
- Forgetting the sign changes with a negative term. In \((1-3x)^4\), every odd power of \(-3x\) is itself negative - it's easy to expand as if \(b=3x\) and lose the alternating signs.
- Dropping the coefficient's own power. In \((2+x)^6\) the term \(2^{6-r}\) must be raised to a power too, not left as a bare 2 - forgetting this is a very common slip when \(a\neq1\).
- Stopping the expansion too early. A question asking for a full expansion needs every term from \(r=0\) to \(r=n\), not just the first two or three - re-read what's being asked before writing a final answer.
Ready to practise properly?
14 binomial-expansion questions, marked instantly like the real exam.
Quick answers
How do I expand \((a+b)^n\)?
Use the binomial theorem \((a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\). The coefficients \(\binom{n}{r}\) can also be read off row \(n\) of Pascal's triangle.
Do I need to memorise the binomial expansion formula?
No, it's given in the formula booklet. You do need to be able to apply it correctly, including handling negative or non-1 values of \(a\) and \(b\). For a GDC-based check, see the parent topic's GDC guidance.