2011/037) If α is a root of equation x² + x + 1 = 0 then show that α is a root of equation x^(3p) + x^(3q+1) + x^(3m+ 2) = 0

Proof:
α is a root of equation x² + x + 1 = 0

so α^2 + α + 1 = 0 and hence α^3 =1

now there are 2 approaches

1)

α ^(3p) + α ^(3q+1) + α ^(3m+ 2)
= (α ^3)^p + (α ^3)^3q* α +( α ^3)3m* α ^2
= 1 + α+ α^2
= 0
Hence α is a root

2)

α ^(3p) + α ^(3q+1) + α ^(3m+ 2)
= α ^(3p) + α ^(3p+1) + α ^(3p+ 2) + α ^(3q+1) - α ^(3p+ 1) + α ^(3m+ 2) - α ^(3p+ 2)
= α ^(3p)( 1+ α + α ^ 2) + α (α ^(3q) - α ^(3p)) + + α^2 (α ^(3m) - α ^(3p))
= 0 + 0+ 0 = 0 as each part is zero
Hence α is a root


as a corollary we see that x² + x + 1 devides x^(3p) + x^(3q+1) + x^(3m+ 2)