# How do you create a polynomial with Degree 4, 2 Positive real zeros, 0 negative real zeros, 2 complex zeros?

Nov 26, 2015

Any polynomial which factors to the form
$c \left(x - a\right) \left(x - b\right) \left(x - \mathrm{di}\right) \left(x + \mathrm{di}\right)$ with $a , b , c , d \in {\mathbb{R}}^{+}$
will satisfy the given conditions.

#### Explanation:

A simple way to creative a polynomial with certain desired zeros is to write it first in its factored form. For example, in this case, let's make a polynomial with the zeros $1 , 2 , i , - i$.

Note that the solutions to
$\left(x - 1\right) \left(x - 2\right) \left(x - i\right) \left(x + i\right) = 0$
are $1 , 2 , i , - i$

Thus we know that $\left(x - 1\right) \left(x - 2\right) \left(x - i\right) \left(x + i\right)$ is a polynomial with the desired properties. Then, all that remains is to multiply it out.

$\left(x - 1\right) \left(x - 2\right) \left(x - i\right) \left(x + i\right) = \left(x - 1\right) \left(x - 2\right) \left({x}^{2} + 1\right)$

$= \left(x - 1\right) \left({x}^{3} - 2 {x}^{2} + x - 2\right)$

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

While the above works, $1 , 2 , i , - i$ were chosen arbitrarily. Any polynomial which factors to the form
$c \left(x - a\right) \left(x - b\right) \left(x - \mathrm{di}\right) \left(x + \mathrm{di}\right)$ with $a , b , c , d \in {\mathbb{R}}^{+}$ will satisfy the given conditions.