Roots of Polynomial

Question

Find a polynomial p∈Z[x] of the lowest degree with leading coefficient 1 such that

p(1+∛2) = 0.

Determine all roots of p.

Answer 1

p(x) = x³ - 3x² + 3x - 3

p(1 + 2^1/3) = (1 + 2^1/3)³ - 3(1 + 2^1/3)² + 3(1 + 2^1/3) - 3

= (1 + 3(2^1/3) + 3(2^2/3) + 2) - 3(1 + 2(2^1/3) + 2^2/3) + 3(1 + 2^1/3) - 3

= 0

Answer 2

p(x) = ( x - (1+2^1/3))*(x-[0.5*(2^1/3 + 2^1/3*3^1/2*i )])*(x-[0.5*(2^1/3 - 2^1/3*3^1/2*i )])

The roots are the expressions subtracted from x in each of the three parentheses sets.