Find a polynomial of lowest degree with rational coefficients that has the given numbers as some of its zeros? -4i, 5

My teacher did a terrible job explaining and I was hoping I could get a better explanation of how to work these kinds of problems out.

Feb 25, 2018

${x}^{3} - 5 {x}^{2} + 16 x - 80$

Explanation:

$5 , 4 i , - 4 i$

So

$\left(x - 4 i\right) \left(x + 4 i\right) \left(x - 5\right) \text{ }$Subtract all from $x$

$\left({x}^{2} - 4 i x + 4 i x - 16 {i}^{2}\right) \left(x - 5\right) \text{ }$ FOIL the first two paratheses

$\left({x}^{2} - 16 {i}^{2}\right) \left(x - 5\right) \text{ }$ Cancel the $4 i$ terms

$\left({x}^{2} + 16\right) \left(x - 5\right) \text{ } {i}^{2} = - 1$, so multiply $- 16$ by $- 1$

${x}^{3} - 5 {x}^{2} + 16 x - 80 \text{ }$ FOIL the remaining paratheses