# How do you write a polynomial function of least degree given the zeros -1, 2i?

Nov 15, 2016

Please see the explanation

#### Explanation:

The zero at -1 implies that (x + 1) is a factor:

$y = \left(x + 1\right)$

The zero at 2i implies that (x - 2i) is a factor.

$y = \left(x + 1\right) \left(x + 2 i\right)$

The zero at 2i implies that -2i is, also, a zero and, therefore, (x + 2i) is a factor.

$y = \left(x + 1\right) \left(x + 2 i\right) \left(x - 2 i\right)$

We know that the product of complex conjugates $a \pm b i$ always produce ${a}^{2} + {b}^{2}$

$y = \left(x + 1\right) \left({x}^{2} + 4\right)$

Use the F.O.I.L method to multiply:

$y = {x}^{3} + 4 x + {x}^{2} + 4$

Rearrange in descending power:

$y = {x}^{3} + {x}^{2} + 4 x + 4$