Find a cubic equation with integral coefficients that has the given roots.
2 and 4+i
since 4+i is a root, its conjugate, 4-i must also be a root. So you have 3 roots, which means 3 factors:
(x - 2) (x - 4 - i) (x - 4 + i)
multiply these out and you have your cubic equation:
y = (x-2)(x² - 8x - 16)
y = x³ -10x² + 32
that's it!
Comments
since 4+i is a root, its conjugate, 4-i must also be a root. So you have 3 roots, which means 3 factors:
(x - 2) (x - 4 - i) (x - 4 + i)
multiply these out and you have your cubic equation:
y = (x-2)(x² - 8x - 16)
y = x³ -10x² + 32
that's it!