The complex number $z$ is chosen so that $1, z, z^2, \, \ldots, z^7$ form the vertices of the regular polygon shown - HSC - SSCE Mathematics Extension 2 - Question 1 - 2017 - Paper 1
Question 1
The complex number $z$ is chosen so that $1, z, z^2, \, \ldots, z^7$ form the vertices of the regular polygon shown.
Which polynomial equation has all of these comp... show full transcript
Worked Solution & Example Answer:The complex number $z$ is chosen so that $1, z, z^2, \, \ldots, z^7$ form the vertices of the regular polygon shown - HSC - SSCE Mathematics Extension 2 - Question 1 - 2017 - Paper 1
Step 1
Determine the vertices of the polygon
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Answer
The complex numbers 1,z,z2,...,z7 correspond to the vertices of a regular octagon in the complex plane, where each vertex can be represented as roots of unity.
Step 2
Identify the polynomial formed by these roots
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Answer
The roots are given by the equation:
z8−1=0
This polynomial represents all the 8th roots of unity.
Step 3
Match with provided options
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Answer
From the options provided, the polynomial that matches our derived polynomial is:
