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The complex number $z$ is chosen so that $1, z, z^2, ext{...}, z^7$ form the vertices of the regular polygon shown - HSC - SSCE Mathematics Extension 2 - Question 1 - 2017 - Paper 1

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The-complex-number-$z$-is-chosen-so-that-$1,-z,-z^2,--ext{...},-z^7$-form-the-vertices-of-the-regular-polygon-shown-HSC-SSCE Mathematics Extension 2-Question 1-2017-Paper 1.png

The complex number $z$ is chosen so that $1, z, z^2, ext{...}, z^7$ 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 $1, z, z^2, ext{...}, z^7$ form the vertices of the regular polygon shown - HSC - SSCE Mathematics Extension 2 - Question 1 - 2017 - Paper 1

Step 1

Identify the regular polygon vertices

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Answer

The vertices of the polygon are represented by the complex numbers 1,z,z2,z3,z4,z5,z6,z71, z, z^2, z^3, z^4, z^5, z^6, z^7. This indicates that the numbers represent the eighth roots of unity. The general form for the nn-th roots of unity can be expressed as: z_k = e^{2 rac{ ext{pi}i}{n}k} for k=0,1,...,7k = 0, 1, ..., 7 when n=8n = 8.

Step 2

Determine the corresponding polynomial equation

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Answer

The polynomial that has these roots can be derived from the fact that the roots of unity satisfy the equation: z8−1=0z^8 - 1 = 0 This can be factored to find its roots at z=1,z,z2,z3,z4,z5,z6,z7z = 1, z, z^2, z^3, z^4, z^5, z^6, z^7. Therefore, it matches the options given in the question.

Step 3

Choose the correct polynomial from the options

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Answer

As per the polynomial derived, the only one that matches is: C. z8−1=0z^8 - 1 = 0.

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