Complex Polynomial Roots

00:00 6 marks

Consider the polynomial \(P(z) = z^3 + (1-2\mathrm{i})z^2 + (-1-2\mathrm{i})z – 1\).

(a) Show that \(z = 1\) is not a root of \(P(z) = 0\). [2]

(b) Given that \(z = \mathrm{i}\) is a root of \(P(z) = 0\), find the other roots of the equation. [4]

Part a

3 minutes 2 marks

Question

Show that \(z = 1\) is not a root of \(P(z) = 0\). [2]

Hint Hint

Hint

Substitute \(z=1\) into \(P(z)\) and simplify to check if the result is zero.

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Complete Answer

Substituting \(z=1\) gives \(P(1)=1+(1-2\mathrm{i})-1-2\mathrm{i}-1=-4\mathrm{i}\), which is not zero; hence, \(z=1\) is not a root.

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Part b

7 minutes 4 marks

Question

Given that \(z = \mathrm{i}\) is a root of \(P(z) = 0\), find the other roots of the equation. [4]

Hint Hint

Hint

Divide \(P(z)\) by \((z-\mathrm{i})\) to get the quadratic \(z^2+(1-\mathrm{i})z+i=0\), then use the quadratic formula.

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Complete Answer

The other roots are \(z = \frac{-1+\sqrt{3}}{2} + \mathrm{i}\frac{1-\sqrt{3}}{2}\) and \(z = \frac{-1-\sqrt{3}}{2} + \mathrm{i}\frac{1+\sqrt{3}}{2}\).

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