# How do you write a polynomial equation of least degree given the roots -3, -2i, 2i?

Apr 18, 2017

The polynomial is ${x}^{3} + 3 {x}^{2} + 4 x + 8$

#### Explanation:

A polynomial equation of least degree given the roots $\alpha , \beta$ and $\gamma$ is

$\left(x - \alpha\right) \left(x - \beta\right) \left(x - \gamma\right)$

Hence, a polynomial equation of least degree given the roots $- 3 , - 2 i$ and $2 i$ is

$\left(x - \left(- 3\right)\right) \left(x - \left(- 2 i\right)\right) \left(x - 2 i\right)$

= $\left(x + 3\right) \left(x + 2 i\right) \left(x - 2 i\right)$

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

= $\left(x + 3\right) \left({x}^{2} - 4 {i}^{2}\right)$

= (x+3)(x^2-4×(-1))

= $\left(x + 3\right) \left({x}^{2} + 4\right)$

= ${x}^{3} + 3 {x}^{2} + 4 x + 12$