Specialist Maths — Question 18

QCAA 2021, Paper 2 · 6 marks

Q18 · 2021 · Technology-activeComplex unfamiliar6 marks

QUESTION 18 (6 marks)

Consider the polynomial

P(z)=z^3+az^2+bz+c

, where

a,b,c\in\mathbb R

and

z\in\mathbb C

. Two of the roots of

P(z)

are also roots of

z^4+z^3+z^2+z+1

. The remaining root of

P(z)

z=2

. Given

z^5-1=(z-1)(z^4+z^3+z^2+z+1)

, determine a possible expression for

P(z)

. Leave your answer in expanded form.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

P(z)=z^3-\dfrac{3+\sqrt5}{2}z^2+\sqrt5z-2

is one possible expression (the conjugate pair at arguments

\pm2\pi/5

Worked solution

The non-real fifth roots of unity occur in conjugate pairs. Taking the pair

\operatorname{cis}(\pm2\pi/5)

gives the quadratic factor

z^2-2\cos(2\pi/5)z+1

. With the remaining root 2,

P(z)=(z-2)\left(z^2-2\cos\frac{2\pi}{5}z+1\right)

=z^3-\frac{3+\sqrt5}{2}z^2+\sqrt5z-2.

The other conjugate pair

\operatorname{cis}(\pm4\pi/5)

gives the other acceptable polynomial.

correctly determines the roots of

z^5=1

[1 mark]

correctly recognises one possible pair of roots

[1 mark]

determines a quadratic factor of

P(z)

in factorised form

[1 mark]

expresses determined quadratic factor of

P(z)

in expanded form

[1 mark]

uses the factor of

z=2

to express

P(z)

in factorised form

[1 mark]

determines

P(z)

in expanded form

[1 mark]

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Specialist Maths — Question 18

QUESTION 18 (6 marks)

QCAA guide · typeset solution

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