# How do you find a third degree polynomial given roots 6 and 3-2i?

Jul 12, 2017

$f \left(x\right) = {x}^{3} - 12 {x}^{2} + 49 x - 78$

#### Explanation:

$\text{note that}$

$\text{complex roots occur in conjugate pairs}$

$3 - 2 i \text{ is a root " rArr3+2i" is also a root}$

$\Rightarrow f \left(x\right) = \left(x - 6\right) \left(x - \left(3 - 2 i\right)\right) \left(x - \left(3 + 2 i\right)\right)$

color(white)(rArrf(x))=(x-6)((x-3)+2i))((x-3)-2i))

$\textcolor{w h i t e}{\Rightarrow f \left(x\right)} = \left(x - 6\right) \left({\left(x - 3\right)}^{2} - 4 {i}^{2}\right)$

$\textcolor{w h i t e}{\Rightarrow f \left(x\right)} = \left(x - 6\right) \left({x}^{2} - 6 x + 13\right)$

$\textcolor{w h i t e}{\Rightarrow f \left(x\right)} = {x}^{3} - 12 {x}^{2} + 49 x - 78$