2023HSCShort answer

Q12 (e) (i)

1 mark

Question

The complex number $2+i$ is a zero of $P(z)=z^4-3z^3+cz^2+dz-30$ , where $c,d\in\mathbb R$ . Explain why $2-i$ is also a zero of $P(z)$ .

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Solution

Criteria	Marks
Provides correct explanation	1

Sample answer:

Since $P(z)$ has real coefficients, zeros occur in complex conjugate pairs.

$\therefore 2+i\text{ a zero}\Rightarrow\overline{2+i}=2-i\text{ is also a zero}$

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Syllabus alignment

Complex Numbers

Solving equations with complex numbersUse De Moivre's theorem and solve equations and polynomial problems involving complex and conjugate roots.

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