I was doing a question and i did the first part fine where i had to show that:
(X+Y+Z)(X+wY+[w^2]Z)(X+[w^2]Y+wZ)= X^3+Y^3+Z^3-3XYZ.
where w=exp(i*2pi/3) i.e. the primitive cube root of unity.
Now the second part implies that the above should be used to solve the equatoion:
x^3-6x+6=0. (*)
My first thoughts were to get the simplified expression in the first part to match equation (*) with X essentially representing x. And then you would have roots as we have shown the expression can be rewritten in a fully factorised form..i.e. the roots are easily deduced from this.
However that means we require numbers for Y,Z such that:
YZ=2 and Y^3+Z^3=6
And i cant think that any values will satisfy this? Can anyone help