Question

How do you write a polynomial with zeros 8, -i, i and leading coefficient 1?

Answer 1

`color(green)(x^3-8x^2+x-8)`

Explanation:

If the polynomial has zeros `8, -i, and i`
then it has factors `cxx(x-8)(x+i)(x-i)` where `c` is a constant.

Since we are asked for a polynomial with a leading coefficient of `1`,
`c=1` and we can ignore it from here on.

`color(blue)((x+i)(x-i))=x^2-i^2=x^2-(-1)=color(blue)(x^2+1)`
and
`(x-8)color(blue)((x+i)(x-i))=(x-8)xxcolor(blue)(""(x^2+1))=x^3-8x^2+x-8`