Home › Formula booklets › Formulas to memorise IB Math AA HL formulas not in the formula booklet
The exam gives you a formula booklet, but it leaves these 12 out. Learn them by heart. Each has a short worked example: tap “Worked example” to open it.
The IB gives this booklet to schools rather than posting it, so we checked the list against our own copy, not the IB's file. Your teacher has the official booklet. Official information: IB DP Mathematics page .
Algebra
Factor and remainder theorems
\(\displaystyle p(a)\)\(\displaystyle {}=0\)\(\displaystyle {}\iff(x-a)\text{ is a factor};\) \(\displaystyle \text{remainder}=p(a)\)
Use it for: Finding unknown coefficients and factorising cubics. The remainder is for dividing \(p(x)\) by \((x-a)\).
Worked example
Given that \((x-2)\) divides \(q(x)=2x^3+kx^2-7x+6\) exactly, work out \(k\). What is left over when \(q(x)\) is divided by \((x+1)\)?
\(q(2)\)\({}=16+4k-14+6\)\({}=0\)\({}\Rightarrow k\)\({}=-2\) Remainder \(=q(-1)=-2-2+7+6=9\) Answer: \(k=-2\), remainder \(9\)
Partial fractions
\(\displaystyle \frac{px+q}{(x-a)(x-b)}\)\(\displaystyle {}=\frac A{x-a}+\frac B{x-b}\)
Use it for: Integration and series expansions of rational functions.
Worked example
Split \(\frac{3x+4}{(x+3)(x-2)}\) into partial fractions.
\(3x+4=A(x-2)+B(x+3)\) \(x=2\): \(10=5B\Rightarrow B=2\); \(x=-3\): \(-5=-5A\Rightarrow A=1\) Answer: \(\frac1{x+3}+\frac2{x-2}\)
Binomial series with a constant term
\(\displaystyle (a+bx)^n=a^n\Big(1+\frac bax\Big)^n,\) \(\displaystyle \text{valid for }|x|<\Big|\frac ab\Big|\)
Use it for: Expanding \((a+bx)^n\) for negative or fractional \(n\).
Worked example
Expand \((2+x)^{-1}\) up to \(x^2\) and state when it is valid.
\((2+x)^{-1}\)\({}=\frac12\big(1+\frac x2\big)^{-1}\) \(=\frac12\big(1-\frac x2+\frac{x^2}4-\cdots\big)\) Valid for \(\big|\frac x2\big|<1\) Answer: \(\frac12-\frac x4+\frac{x^2}8\), valid for \(|x|<2\)
Complex numbers
nth roots of a complex number
\(\displaystyle z^n=re^{i\theta}\ \)\(\displaystyle {}\Rightarrow\ z=r^{\frac1n}e^{i\frac{\theta+2\pi k}n},\) \(\displaystyle k=0,1,\dots,n-1\)
Use it for: Solving \(z^n=w\): de Moivre gives powers, not the full set of roots.
Worked example
Solve \(z^3=8i\), giving answers in the form \(a+bi\).
\(8i=8e^{i\frac\pi2}\), so \(|z|=2\) Arguments \(\frac\pi6,\ \frac{5\pi}6,\ \frac{3\pi}2\) (adding \(\frac{2\pi}3\) each time) \(2e^{i\frac\pi6}=\sqrt3+i\), \(2e^{i\frac{5\pi}6}\)\({}=-\sqrt3+i\), \(2e^{i\frac{3\pi}2}=-2i\) Answer: \(\sqrt3+i,\ -\sqrt3+i,\ -2i\)
Powers of z on the unit circle
\(\displaystyle |z|=1:\) \(\displaystyle z^n+z^{-n}\)\(\displaystyle {}=2\cos n\theta,\) \(\displaystyle z^n-z^{-n}\)\(\displaystyle {}=2i\sin n\theta\)
Use it for: Writing \(\cos^n\theta\) or \(\sin^n\theta\) in terms of multiple angles.
Worked example
Use \(z=e^{i\theta}\) to show that \(\cos^3\theta\)\({}=\frac14(\cos3\theta+3\cos\theta)\).
Expand: \((z+z^{-1})^3\) \(=z^3+3z+3z^{-1}+z^{-3}\) \((2\cos\theta)^3\)\({}=2\cos3\theta+3(2\cos\theta)\) Divide by 8. Answer: \(\cos^3\theta\)\({}=\frac14\cos3\theta+\frac34\cos\theta\)
Vectors
Area of a triangle from vectors
\(\displaystyle \text{Area}\)\(\displaystyle {}=\tfrac12\big|\overrightarrow{AB}\times\overrightarrow{AC}\big|\)
Use it for: 3D triangle areas (the booklet gives the parallelogram).
Worked example
Points \(P(2,1,0)\), \(Q(4,3,1)\) and \(R(1,0,3)\) form a triangle. Work out its area.
\(\overrightarrow{PQ}\)\({}=(2,2,1)\), \(\overrightarrow{PR}\)\({}=(-1,-1,3)\) \(\overrightarrow{PQ}\times\overrightarrow{PR}\)\({}=(7,-7,0)\), magnitude \(7\sqrt2\) Answer: \(\frac{7\sqrt2}2\)\({}\approx4.95\)
Calculus
Implicit differentiation
\(\displaystyle \frac{d}{dx}f(y)=f'(y)\frac{dy}{dx},\) \(\displaystyle \frac{d}{dx}(xy)=x\frac{dy}{dx}+y\)
Use it for: Gradients of curves you cannot write as \(y=\ldots\).
Worked example
Find \(\frac{dy}{dx}\) at \((1,2)\) on \(x^2+xy+y^2=7\).
\(2x+y+x\frac{dy}{dx}+2y\frac{dy}{dx}\)\({}=0\) \(\frac{dy}{dx}\)\({}=-\frac{2x+y}{x+2y}\)\({}=-\frac{4}{5}\) at \((1,2)\) Answer: \(-\frac45\)
L'Hôpital's rule
\(\displaystyle \lim_{x\to a}\frac{f(x)}{g(x)}\)\(\displaystyle {}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\) \(\displaystyle \text{for }\tfrac00\text{ or }\tfrac\infty\infty\)
Use it for: Limits that give \(\frac00\).
Worked example
Find \(\lim_{x\to0}\frac{e^{2x}-1}{\sin3x}\).
Direct substitution gives \(\frac00\). Differentiate top and bottom: \(\frac{2e^{2x}}{3\cos3x}\to\frac23\) Answer: \(\frac23\)
Integrating sin² and cos²
\(\displaystyle \sin^2x\)\(\displaystyle {}=\tfrac12(1-\cos2x),\) \(\displaystyle \cos^2x\)\(\displaystyle {}=\tfrac12(1+\cos2x)\)
Use it for: Integrating even powers of sin and cos.
Worked example
Find \(\int_0^{\frac\pi2}\sin^2x\,dx\).
\(\int_0^{\frac\pi2}\tfrac12(1-\cos2x)\,dx\) \(=\big[\tfrac x2-\tfrac{\sin2x}4\big]_0^{\frac\pi2}\)\({}=\tfrac\pi4\) Answer: \(\frac\pi4\)
Integrating f'(x)/f(x)
\(\displaystyle \int\frac{f'(x)}{f(x)}\,dx\)\(\displaystyle {}=\ln|f(x)|+C\)
Use it for: Any fraction whose top is (a multiple of) the derivative of the bottom.
Worked example
Find \(\int_0^2\frac{2x}{x^2+5}\,dx\).
The top is the derivative of \(x^2+5\). \(\big[\ln(x^2+5)\big]_0^2\)\({}=\ln9-\ln5\) Answer: \(\ln\frac95\)
Homogeneous differential equations
\(\displaystyle \frac{dy}{dx}=f\Big(\frac yx\Big):\) \(\displaystyle y=vx,\) \(\displaystyle \frac{dy}{dx}=v+x\frac{dv}{dx}\)
Use it for: First-order equations where every term has the same degree.
Worked example
Solve \(\frac{dy}{dx}=1+\frac yx\), \(x>0\), given \(y=2\) when \(x=1\).
\(v+x\frac{dv}{dx}\)\({}=1+v\)\({}\Rightarrow\frac{dv}{dx}\)\({}=\frac1x\) \(v\)\({}=\ln x+C\)\({}\Rightarrow y=x\ln x+Cx\) \(x=1,\ y=2\Rightarrow C=2\) Answer: \(y=x\ln x+2x\)
Probability
Median of a continuous random variable
\(\displaystyle \int_{-\infty}^{m}f(x)\,dx\)\(\displaystyle {}=\frac12\)
Use it for: Continuous random variable questions (the mode is the maximum of \(f\)).
Worked example
\(f(x)=\frac38x^2\) for \(0\le x\le2\). Find the median.
\(\int_0^m\frac38x^2\,dx\)\({}=\frac{m^3}8\)\({}=\frac12\) \(m^3=4\) Answer: \(m=\sqrt[3]4\approx1.59\)
Questions
Which IB Math AA HL formulas are not in the formula booklet? The ones to learn by heart are: Factor and remainder theorems; Partial fractions; Binomial series with a constant term; nth roots of a complex number; Powers of z on the unit circle; Area of a triangle from vectors; Implicit differentiation; L'Hôpital's rule; Integrating sin² and cos²; Integrating f'(x)/f(x); Homogeneous differential equations; Median of a continuous random variable. Each one is on this page with a worked example.
Do I get a formula booklet in the exam? Yes. You get a clean copy of the IB formula booklet in every IB Mathematics exam, so you only need to memorise what it leaves out.
What is the best way to memorise them? Cover the formula, write it from memory, then try the worked example without looking. Come back to the ones you missed the next day. Our flashcards do the spacing for you.
Is this page a copy of the official booklet? No. We link to the official IB Mathematics: analysis and approaches formula booklet; we do not reproduce it. This page lists only what the booklet leaves out, in our own words, with our own examples.
Our own list, wording and examples, written and checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.