complex numbers question from cambridge (1 Viewer)

[azn5reppin]

Ex 2.4 q6 is "factorise z^3-1. If z is one of the cube roots of unity, find the two possible values of z^2+z+1. " the solutions do z=1, z^2+z+1=3. Z not equal to 1, z^ +z+1=0. I dont rly understand how the roots of unity relate to the question, can someone explain?

 

[Kurosaki]

Is the root of unity when you have like z^n=1?

 

[azn5reppin]

Is the root of unity when you have like z^n=1?

Think so. Or z^n=-1? Not too sure

 

[bottleofyarn]

Think so. Or z^n=-1? Not too sure

Roots of unity are when
So in the question, you would relate it to and thus find the values for

 

[braintic]

Spot on, roots of unity are when
So in the question, you would relate it to and thus find the values for

Since when is -1 considered to be 'the unit' ?

Roots of unity are only the solutions to z^n = 1.

If you want to extend this to -1, why not i, -i, and any other number with a modulus of 1?

 

[bottleofyarn]

Since when is -1 considered to be 'the unit' ?

Roots of unity are only the solutions to z^n = 1.

If you want to extend this to -1, why not i, -i, and any other number with a modulus of 1?

Brain fart. I'll edit cause braintic is 100% correct.

 

[azn5reppin]

So is z^n=1 or -1 the only type of 'roots of unity' questions?

 

[bottleofyarn]

Roots of unity is only for . Roots of unity questions are all based on this and typically use factorisation or de Moivre's theorem. Sorry for any earlier confusion.

 

[Fizzy_Cyst]

just z^n=1

 

[azn5reppin]

So would you solve these roots of unity questions different to any other question like find fourth roots of 2+2i? And sorry I still dont understand how roots of unity relates to the original question in my first post, can someone explain?

 

[obliviousninja]

z^3-1=0

(z-1)(z^2+z+1)=0

use quadratic formula

 

[anomalousdecay]

So would you solve these roots of unity questions different to any other question like find fourth roots of 2+2i? And sorry I still dont understand how roots of unity relates to the original question in my first post, can someone explain?

To find the fourth roots of something like 2+2i, you must find its equivalence in mod-arg form, then use De Moivre's theorem.

Another quick thing to do is let:

 

[braintic]

Ex 2.4 q6 is "factorise z^3-1. If z is one of the cube roots of unity, find the two possible values of z^2+z+1. " the solutions do z=1, z^2+z+1=3. Z not equal to 1, z^ +z+1=0. I dont rly understand how the roots of unity relate to the question, can someone explain?

As no-one has actually answered the original question:

This question does NOT require either de Moivres theorem OR the quadratic formula. It is MUCH more trivial.

z^3 - 1 = 0

Factorise:
(z-1) (z^2 + z + 1) = 0

So

EITHER z=1 making z^2+z+1=1+1+1=3

OR z^2 + z + 1 = 0

Done.