Question

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.

Answer 1

`x^3-5x^2+16x-80`

Explanation:

`5, 4i, -4i`

So

`(x-4i)(x+4i)(x-5) " "`Subtract all from `x`

`(x^2-4ix+4ix-16i^2)(x-5) " "` FOIL the first two paratheses

`(x^2-16i^2)(x-5)" "` Cancel the `4i` terms

`(x^2+16)(x-5) " "i^2 =-1`, so multiply `-16` by `-1`

`x^3-5x^2+16x-80" "` FOIL the remaining paratheses