# How do you find a polynomial function of degree 3 with 5, i, -i as zeros?

The best way is to recognise that, if $x = 5$ is a root, then $x - 5 = 0$, and ditto for the other two roots. So we have $x - 5 , x - i , x + i$ all equalling zero. To find our polynomial, we just multiply the three terms together:
$\left(x - 5\right) \left(x - i\right) \left(x + i\right)$
$= \left({x}^{2} - i x - 5 x + 5 i\right) \left(x + i\right)$
$= {x}^{3} + i {x}^{2} - i {x}^{2} - \left({i}^{2}\right) x - 5 {x}^{2} - 5 i x + 5 i x + 5 {i}^{2}$
${x}^{3} - 5 {x}^{2} + x - 5$.