Solving quadratic, cubic and quartic equations with complex roots, arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem and loci.

What this dot point is asking

AQA wants you to work fluently with complex numbers in Cartesian, modulus-argument and exponential form, plot them on an Argand diagram, use de Moivre's theorem to find powers and roots, solve polynomial equations with complex roots, and sketch loci defined by modulus and argument conditions.

Arithmetic and the conjugate

For $z = x + yi$ the complex conjugate is $z^* = x - yi$. Multiplying gives $zz^* = x^2 + y^2 = |z|^2$, which is always real and non-negative. To divide, multiply numerator and denominator by the conjugate of the denominator.

The Argand diagram and modulus-argument form

The modulus is $r = |z| = \sqrt{x^2 + y^2}$ and the argument $\theta = \arg z$ is the angle from the positive real axis, measured in $(-\pi, \pi]$. Always sketch the point first so you pick the correct quadrant for $\theta$.

De Moivre's theorem

For any integer $n$, $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$. This is the engine for powers, for the $n$th roots of a complex number, and for deriving multiple-angle identities such as $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$.

Roots of polynomials

For a polynomial with real coefficients, complex roots occur in conjugate pairs. If $2 + 3i$ is a root then so is $2 - 3i$, and their sum and product give a real quadratic factor.

Roots of unity and the nth roots of a complex number

To find all $n$th roots of a complex number $w = R(\cos\phi + i\sin\phi)$, write $w$ in modulus-argument form, take the real $n$th root of the modulus, and divide the argument (with all its $2\pi k$ equivalents) by $n$. The $n$ roots are

They all share the modulus $R^{1/n}$ and are equally spaced by $\frac{2\pi}{n}$ around a circle, so on an Argand diagram they form the vertices of a regular $n$-gon. The $n$th roots of unity in particular are powers of $\omega = e^{2\pi i/n}$, and they sum to zero because they are the roots of $z^n - 1 = 0$, whose $z^{n-1}$ coefficient is zero. This geometric picture is examined directly: AQA often asks you to mark the roots on a diagram and comment on their symmetry.

Loci on the Argand diagram

Conditions on $z$ describe curves, and the marks come from translating the algebra into a labelled sketch. $|z - a| = r$ is a circle of radius $r$ centred at the point $a$, because $|z - a|$ is the distance from $z$ to $a$. $|z - a| = |z - b|$ is the perpendicular bisector of the segment joining $a$ and $b$, the set of points equidistant from the two. $\arg(z - a) = \theta$ is a half-line (ray) starting at $a$ and making angle $\theta$ with the positive real direction, with the start point $a$ itself excluded since $\arg 0$ is undefined. Inequalities such as $|z - a| \leq r$ shade a region (here the closed disc), and intersections of two loci pick out the points or arcs satisfying both conditions at once.

These tools combine on every paper: arithmetic and the conjugate for division, de Moivre for powers and roots, the conjugate-pair fact for polynomials, and the Argand diagram for loci and the geometry of roots of unity.