Binomial Expansions with Unknowns (AA HL)
Most binomial questions give you every number up front, but this group hides one - an unknown power \(n\), an unknown coefficient inside the bracket, or a condition linking two terms - and asks you to recover it algebraically. The expansion itself works exactly as normal; the extra step is turning the given condition into an equation and solving it. It's part of the broader Binomial Theorem topic.
30 questions on this sub-topic.
The formulas you need
Covered under IB syllabus reference SL1.9: the binomial theorem for expansion of \((a+b)^n\) with \(n \in \mathbb{N}\), using Pascal's triangle or \(\binom{n}{r}\), found by formula or by technology. Both formulas below are in the formula booklet.
Binomial theorem
\((a+b)^n=\displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)
Positive integer \(n\). When \(n\) is itself the unknown, leave it as a letter in this formula and build an equation from whatever condition the question gives you.
Binomial coefficient
\(\binom{n}{r}=\dfrac{n!}{r!(n-r)!}\)
Use this to compare or equate two coefficients symbolically - the ratio \(\binom{n}{r+1}/\binom{n}{r}\) often simplifies an "equal coefficients" condition down to a single linear equation.
Need the full syllabus wording and formula-booklet reference table? See Binomial Theorem. GDC methods for these expansions are covered in the parent topic's GDC section.
Worked examples
Find the coefficient of \(x^4\) in \((1+x+x^2)(1+x)^6.\)
Worked solution
\(x^4:15,\ x^3:20,\ x^2:15.\) M1 A1
(each factor of \(1, x, x^2\) shifts the power): \(1\cdot 15 + 1\cdot 20 + 1\cdot 15\) M1 \(= 50.\) A1
In the expansion of \((1 + x)^{n}\), the coefficients of \(x^4\) and \(x^5\) are equal.
Find \(n.\)
Worked solution
\(\binom n4 = \binom n5.\) M1
\(\dfrac{\binom n5}{\binom n4} = \dfrac{n-4}{5}.\) M1 A1
\(\dfrac{n-4}{5} = 1 \Rightarrow n - 4 = 5 \Rightarrow n = 9.\) A1
Expand \((2-x)^4\) fully.
Worked solution
\(\binom40 2^4 - \binom41 2^3 x + \binom42 2^2 x^2 - \binom43 2x^3 + \binom44 x^4.\) M1
\(16 - 32x + 24x^2 - 8x^3 + x^4.\) A1 A1 A1
Common mistakes
- Ignoring the validity range \(|x|<1\). If the question extends to a fractional or negative power, the expansion only converges where \(|x|<1\) - state this range whenever asked to justify an approximation.
- Assuming an unknown \(n\) fixes the position \(r\) too. When a condition links two coefficients, \(n\) and \(r\) are two separate unknowns - write both binomial coefficients in terms of both letters before you start cancelling.
- Expanding \(\binom{n}{r}\) from scratch instead of using the ratio. Full factorial expansion of an unknown-\(n\) coefficient gets messy fast; forming \(\binom{n}{r+1}/\binom{n}{r}\) collapses almost everything and leaves one clean linear equation.
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30 binomial-with-unknowns questions, marked instantly like the real exam.
Quick answers
How do I expand a binomial like \((1+x)^n\)?
Use the binomial theorem \((a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\), reading the coefficients \(\binom{n}{r}\) off Pascal's triangle or from the formula \(\dfrac{n!}{r!(n-r)!}\).
What if the power \(n\) is unknown in the question?
Keep \(n\) as a letter in the general term, turn whatever condition the question gives (two coefficients equal, one coefficient stated) into an equation, and solve that equation for \(n\).