Define the argument of z.

The angle between the positive real axis and the line from the origin to z, measured anticlockwise.

How do you divide two complex numbers?

Multiply top and bottom by the conjugate of the denominator.

When are two complex numbers equal?

When their real parts are equal and their imaginary parts are equal.

State the conjugate root theorem.

If a polynomial with real coefficients has root z1, then z1^ is also a root.

How many real roots can a real cubic have?

One or three; non-real roots come in pairs.

Method to solve a quartic given one complex root?

Use the conjugate to form a real quadratic factor, divide, then solve the remaining quadratic.

Why can the conjugate root theorem fail?

It needs all coefficients real; a complex coefficient breaks the pairing.

Complex numbers and quadratic, cubic and quartic equationsEdexcel A-Level Further Maths: Flashcards

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What is the imaginary part of $3-5i$?

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What is the imaginary part of $3-5i$ ?: $-5$ (a real number, not $-5i$ ).
Define the modulus of $z=x+iy$ .: $|z|=\sqrt{x^2+y^2}$ .
Define the argument of $z$ .: The angle between the positive real axis and the line from the origin to $z$ , measured anticlockwise.
What is the complex conjugate of $x+iy$ ?: $z^*=x-iy$ .
What is $zz^*$ ?: $x^2+y^2=|z|^2$ , which is real.
How do you divide two complex numbers?: Multiply top and bottom by the conjugate of the denominator.
When are two complex numbers equal?: When their real parts are equal and their imaginary parts are equal.
State the conjugate root theorem.: If a polynomial with real coefficients has root $z_1$ , then $z_1^*$ is also a root.
Real quadratic factor from the roots $p\pm qi$ ?: $z^2-2pz+(p^2+q^2)$
Roots of $z^2-4z+13=0$ ?: $z=2\pm3i$
How many real roots can a real cubic have?: One or three; non-real roots come in pairs.
Method to solve a quartic given one complex root?: Use the conjugate to form a real quadratic factor, divide, then solve the remaining quadratic.
Why can the conjugate root theorem fail?: It needs all coefficients real; a complex coefficient breaks the pairing.

Exam questions on Complex numbers and quadratic, cubic and quartic equations

The complex numbers $z_1=3+4i$ and $z_2=1-2i$ are given.
Find the exact value of $|z_1z_2|$ .2 marks
The complex number $z=x+iy$ , where $x$ and $y$ are real, satisfies $z+2z^*=9-2i$ .
Hence find the modulus of $z$ and its argument, in radians to 2 decimal places.2 marks
The quartic equation $z^4-8z^3+27z^2-50z+50=0$ has $1+2i$ as one root. All of its coefficients are real.
Write down another root of the equation and hence find a quadratic factor with real coefficients.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).