Binomial With Unknowns (AA SL)
These questions flip the usual binomial problem around: instead of a fully-defined expansion, you're given information about a coefficient and asked to find the missing power \(n\), constant \(a\), or parameter \(k\) that produced it. The result is usually an equation - sometimes a quadratic - to solve rather than a straightforward substitution. It's part of the broader Binomial Theorem topic.
21 questions on this sub-topic.
The formulas
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 binomial expansion
\((a+b)^n=\displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)
Write out the coefficient of the term you're told about - in terms of the unknown - before setting up any equation.
✓ In the formula bookletBinomial coefficient \(nC_r\)
\(\displaystyle\binom{n}{r}=\dfrac{n!}{r!\,(n-r)!}\)
When \(n\) itself is unknown, this expands to a product of consecutive integers over \(r!\) - useful for turning "coefficient of \(x^2\)" into an equation in \(n\).
✓ In the formula bookletNeed the full syllabus wording and formula-booklet reference table? See Binomial Theorem.
Worked examples
In the expansion of \((1+x)^n\), the coefficient of \(x^2\) is 15.
Find \(n\).
Worked solution
In \((1+x)^n\) the coefficient of \(x^2\) is \(\binom{n}{2}\); set it equal to 15. M1
\(\dfrac{n(n-1)}{2}=15\Rightarrow n(n-1)=30.\) A1A1
\(n^2-n-30=0\Rightarrow(n-6)(n+5)=0\), and since \(n>0\), \(n=6.\) A1
In the expansion of \((1+kx)^5\), the coefficient of \(x^2\) is 40.
(a)(i) Find the value of \(k<0\).
(a)(ii) Find the value of \(k>0\).
Worked solution
The coefficient of \(x^2\) in \((1+kx)^5\) is \(\binom{5}{2}k^2\), because each \(x\) brings a factor \(k\). M1
Setting this equal to 40: \(10k^2=40\Rightarrow k^2=4.\) A1
So \(k=\pm2\) (both valid, no domain restriction). A1A1
Common mistakes
- Rejecting solutions without checking which are actually invalid. A quadratic in \(n\) often gives one positive and one negative root - since \(n\) must be a positive whole number, only the negative root is usually rejected, but check the question doesn't restrict \(k\) or \(a\) in the same way before discarding a solution.
- Losing the sign when squaring \(k\) or \(a\). An equation like \(k^2=4\) has two roots, \(k=2\) and \(k=-2\) - dropping the negative solution (or the positive one) loses marks if the question asks for both, or specifies which one it wants.
- Substituting the wrong power into the binomial coefficient. For the coefficient of \(x^2\), it's \(\binom{n}{2}\) that's needed, not \(\binom{n}{n-2}\) or \(\binom{2}{n}\) - although the first two happen to be equal, mixing up the order of \(n\) and \(r\) is a frequent slip when \(n\) itself is the unknown.
Ready to practise properly?
21 questions on binomial expansions with unknowns, marked instantly like the real exam.
Quick answers
How do I find \(n\) if I'm told the coefficient of a term in a binomial expansion?
Write the coefficient using \(\binom{n}{r}\) in terms of \(n\), set it equal to the given value, and solve the resulting equation - often a quadratic in \(n\) - by factorising or the quadratic formula, then reject any negative or non-integer solution.
Why do I sometimes get two values for \(k\) or \(a\)?
Because the coefficient involves \(k^2\) or \(a^2\), squaring loses the sign - both the positive and negative square root can genuinely satisfy the original equation unless the question restricts the value, e.g. \(k>0\). See the parent topic's GDC guidance for checking values of \(nC_r\).