State the conjugate root theorem.

If a polynomial has real coefficients, its non-real roots occur in conjugate pairs.

Why does the theorem need real coefficients?

Conjugation changes p(z0)=0 into p(z0^)=0 only if the coefficients are unchanged by conjugation, i.e. real.

Steps to solve a cubic given a real root?

Divide by the linear factor (or compare coefficients), then solve the quadratic.

Steps to solve a quartic given a complex root a+bi?

State a-bi, form z^2-2az+a^2+b^2, find the other quadratic factor, solve both.

Complex conjugates and polynomial rootsAQA A-Level Further Maths: Flashcards

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What is the complex conjugate of $3-2i$?

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What is the complex conjugate of $3-2i$ ?: $3+2i$
What is $zz^*$ for $z=x+iy$ ?: $x^2+y^2$ , which is real.
State the conjugate root theorem.: If a polynomial has real coefficients, its non-real roots occur in conjugate pairs.
Why does the theorem need real coefficients?: Conjugation changes $p(z_0)=0$ into $p(z_0^*)=0$ only if the coefficients are unchanged by conjugation, i.e. real.
Quadratic with roots $a\pm bi$ ?: $z^2-2az+a^2+b^2$
Real quadratic factor with roots $2\pm i$ ?: $z^2-4z+5$
How many real roots must a real cubic have?: At least one: $1$ or $3$ .
How many real roots can a real quartic have?: $0$ , $2$ or $4$ .
Steps to solve a cubic given a real root?: Divide by the linear factor (or compare coefficients), then solve the quadratic.
Steps to solve a quartic given a complex root $a+bi$ ?: State $a-bi$ , form $z^2-2az+a^2+b^2$ , find the other quadratic factor, solve both.
Roots of $z^2+4=0$ ?: $\pm2i$
Solve $z^2-2z+5=0$ .: $z=1\pm2i$

Exam questions on Complex conjugates and polynomial roots

The cubic equation $z^3-5z^2+17z-13=0$ has a real root $z=1$ .
Explain why the cubic has exactly one real root.2 marks
The cubic equation $z^3-4z^2+6z-4=0$ has real coefficients and a root $z=1+i$ .
Show by substitution that $1-i$ is a root of the equation.2 marks
The quartic $f(z)=z^4-2z^3+9z^2-8z+20$ has $z^2-2z+5$ as a factor.
Find the other quadratic factor of $f(z)$ .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).