Another Complex Question about Roots

[KirSky]

I've encounter this question whilst studying for the complex number exam.



I did part A but don't know how to tackle part B.

Does this question require 4U polynomial? Cause my school didn't do it yet.

Thanks!

[RuiAce]

I've encounter this question whilst studying for the complex number exam.

(Image removed from quote.)

I did part A but don't know how to tackle part B.

Does this question require 4U polynomial? Cause my school didn't do it yet.

Thanks!

Preferably with knowledge of polynomials. But at your current level, you should still attempt to understand why the answer is just \(z^3 - 1 = 0\)

Spoiler

Cube root of unity

[KirSky]

I still don't really get why it is z^3-1=0. For roots of unity, the question stated that it does not include alpha tho.

My friend's solution suggests another answer which uses the assumption that conjugate of a complex root is also a root.



He says the equation is z^3+z^2-1=0

I am so confused..

[RuiAce]

Hang on, my bad. I was in a hurry at the time and didn’t read the question properly.

If we don’t want \(\alpha\) to be a root then the question entirely falls under the polynomials topic. This is because we need a theorem involving polynomials whose coefficients are real. (Edit: Which is the theorem your friend suggested.)

[KirSky]

Welp, then my friend is pretty ahead of me I suppose  Good to know this theorem anyways because it's very convenient.