Question

How do you write a polynomial with zeros 8,-i, i?

Answer 1

This polynomial is `(x^3 - 8x^2 + x - 8)`

Explanation:

We can use the factored form of polynomials to write this first, then FOIL it out into the STANDARD FORM.

since the polynomial has 3 given zeroes, we can write

`(x-8)(x+i)(x-i)`
We do this because letting x = 8,-i, or i would mean that the polynomial becomes (with an intended name) zero.

Now we can just distribute it out.

First do the two complex zeroes
`(x+i)(x-i) = x^2 - i^2`
`i^2 = -1`
so `x^2 - i^2` becomes `x^2 + 1`
Now, just FOIL out `(x-8)(x^2+1)`

`(x^3 - 8x^2 + x - 8)`
And we are done.