F.5 M2 about Determinants

27x + y + 3z = p^3
9x - y - 3z = p^2
3x + y + 3z = pthe second and third equations multiplied by 3,
27x+y+3z=p^3
27x-3y-9z=3p^2
27x+9y+27z=9pthe second and third equations both minus the first equation respectively,
27x+y+3z=p^3
-4y-12z=3p^2-p^3
8y+24z=9p-p^3the third equation plus (the second equation multiplied by 2),
27x+y+3z=p^3
-4y-12z=3p^2-p^3
0=9p-p^3+2(3p^2-p^3)So,we have
-3p^3+6p^2+9p=0
-3p(p^2-2p-3)=0
-p(p+1)(p-3)=0
p=0 or p=-1 or p=3