Polynomial Roots and Factorization

00:00 18 marks

Consider the polynomial \(P(z) = z^4 – 2z^3 + 7z^2 – 4z + 10\), where \(z \in \mathbb{C}\).
(a) Given that \(z = 1 + 2\mathrm{i}\) is a root of the equation \(P(z)=0\), write down another complex root. [1]
(b) Find a quadratic factor of \(P(z)\) with real coefficients. [4]
(c) Hence, find all the roots of the equation \(P(z)=0\). [6]
(d) Let the roots be \(z_1, z_2, z_3, z_4\). Find the value of \(z_1^* + z_2^* + z_3^* + z_4^*\). [3]
(e) Find the product of the roots of \(P(z)=0\). [4]

Part a

2 minutes 1 marks

Question

Given that \(z = 1+2\mathrm{i}\) is a root of \(P(z)=0\), write down another complex root.

Hint Hint

Hint

Recall that for polynomials with real coefficients, complex roots occur in conjugate pairs.

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\(1-2\mathrm{i}\)

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Part b

4 minutes 4 marks

Question

Find a quadratic factor of \(P(z)\) with real coefficients.

Hint Hint

Hint

Form the factor by multiplying \((z-(1+2\mathrm{i}))\) and \((z-(1-2\mathrm{i}))\).

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\(z^2-2z+5\)

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Part c

6 minutes 6 marks

Question

Hence, find all the roots of the equation \(P(z)=0\).

Hint Hint

Hint

Divide \(P(z)\) by \(z^2-2z+5\) to obtain the remaining quadratic factor and solve for \(z\).

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The roots are \(1+2\mathrm{i}\), \(1-2\mathrm{i}\), \(\mathrm{i}\sqrt{2}\), and \(-\mathrm{i}\sqrt{2}\).

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Part d

3 minutes 3 marks

Question

Let the roots be \(z_1, z_2, z_3, z_4\). Find the value of \(z_1^* + z_2^* + z_3^* + z_4^*\).

Hint Hint

Hint

Notice that taking the conjugate of each root and summing gives the same as the sum of the original roots.

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Part e

4 minutes 4 marks

Question

Find the product of the roots of \(P(z)=0\).

Hint Hint

Hint

Recall that for a monic polynomial, the product of the roots equals the constant term.

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