Complex numbers & polynomial roots

The topic on one screen

• Why complex numbers matter: they make every non-constant polynomial factor completely. For a polynomial with real coefficients, any non-real root $a+ib$ forces the conjugate $a-ib$ to be a root too (CP-2.1 to CP-2.3).
• In Cartesian form, calculate with $z=x+iy$ and use $i^2=-1$. For division, multiply top and bottom by the conjugate of the denominator. Worked example: $\dfrac{3+4i}{1-2i}=\dfrac{(3+4i)(1+2i)}{5}=-1+2i$ (CP-2.2 to CP-2.3).
• On an Argand diagram, $x=\operatorname{Re}(z)$ and $y=\operatorname{Im}(z)$. The modulus $|z|=\sqrt{x^2+y^2}$ is a distance; the argument is a directed angle. Use $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$, with the quadrant checked before stating $\arg z$ (CP-2.4 to CP-2.6 and CP-2.9).
• Loci are geometry disguised as algebra: $|z-a|=r$ is a circle, $|z-a|=|z-b|$ is a perpendicular bisector, and $\arg(z-a)=\theta$ is a ray from $a$. Strict inequalities exclude the boundary (CP-2.7).
• De Moivre gives $[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)$. For $\theta\notin2\pi\mathbb Z$, start trigonometric sums from $\sum_{r=0}^{n-1}e^{ir\theta}=(1-e^{in\theta})/(1-e^{i\theta})$, then take real or imaginary parts; when $\theta\in2\pi\mathbb Z$, the sum is $n$. Euler form is usually shortest for products, quotients and powers (CP-2.8 to CP-2.9).
• For $z^n=re^{i\theta}$, take modulus $r^{1/n}$ and arguments $\dfrac{\theta+2k\pi}{n}$ for $k=0,1,\ldots,n-1$. Worked example: the fifth roots of $32e^{i\pi}$ have modulus $2$ and arguments $\dfrac{(2k+1)\pi}{5}$; they are vertices of a regular pentagon (CP-2.10).
• For roots of unity, rotation by one vertex is multiplication by $\omega=e^{2\pi i/n}$, and $1+\omega+\cdots+\omega^{n-1}=0$. Use these algebraic facts to prove equal lengths, angles, centroids and other geometry of the regular $n$-gon (CP-2.11).
• For roots $\alpha,\beta,\gamma$ of a monic cubic, compare coefficients: $\sum\alpha=-a$, $\sum\alpha\beta=b$, and $\alpha\beta\gamma=-c$. The signs alternate. To transform roots, write the old root in terms of the new one and substitute into the original polynomial (CP-4.1 to CP-4.2).
• Worked example: if $x^3-3x+1=0$ has roots $\alpha,\beta,\gamma$, put $y=x+2$. Substituting $x=y-2$ gives $y^3-6y^2+9y-1=0$, whose roots are $\alpha+2,\beta+2,\gamma+2$.
• Common errors: giving only one value of an argument, losing roots by using a calculator's principal root, forgetting a conjugate, reversing the transformation substitution, or drawing an Argand locus without an excluded endpoint.
• Exam technique: show modulus and argument equations before listing roots; label every root on a diagram; and when asked to 'show', preserve the exact target form instead of replacing it with a decimal equivalent.

Where students actually lose marks

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Questions are written in the style of past Edexcel papers (source shown on each) — never copied from them.

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