# Complex cube root of 1

1. Feb 5, 2009

### Ferrus

1. The problem statement, all variables and given/known data

If w is a complex cube root of 1, prove that x + wy + w^2z is a factor of x^3+ y^3 + z^3 - 3xyz, and hence factorise the equation completely.

2. Relevant equations

Complex cube root of 1 = -1/2 +/- 3^1/2/2 i

3. The attempt at a solution

Erm, I feel way over my head. I have tried plugging in the equation to the first one but this doesn't seem to generate anything intelligible for me.

2. Feb 5, 2009

### gabbagabbahey

Hi Ferrus, welcome to PF

Hint 1: If $x + wy + w^2z$ is a factor of $x^3+ y^3 + z^3 - 3xyz$, what can you say about the remainder of $$\frac{x^3+ y^3 + z^3 - 3xyz}{x + wy + w^2z}$$

Hint 2: If $z^3=1$, then $z^3-1=(z-1)(z^2+z+1)=0$.... so what can you say about $w^2+w+1$?

3. Feb 5, 2009

### tiny-tim

Welcome to PF!

Hi Ferrus ! Welcome to PF!

Hint: if (x+a) is a factor of a polynomial, then put x = -a and the polynomial will be zero …

so put x = … ?