# How do you find a third degree polynomial given roots -7 and i?

${x}^{3} + 7 {x}^{2} + x + 7$

#### Explanation:

The imaginary or complex roots of a polynomial equation are always conjugate hence if $i$ is a root then $- i$ will also be another root.

Thus, the cubic polynomial $P \left(x\right)$ has three roots $\setminus \alpha = - 7 , \setminus \setminus \beta = i , \setminus \setminus \setminus \gamma = - i$ hence the cubic polynomial is given as

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

$= \left(x - \left(- 7\right)\right) \left(x - i\right) \left(x - \left(- i\right)\right)$

$= \left(x + 7\right) \left(x - i\right) \left(x + i\right)$

$= \left(x + 7\right) \left({x}^{2} - {i}^{2}\right)$

$= \left(x + 7\right) \left({x}^{2} + 1\right)$

$= {x}^{3} + 7 {x}^{2} + x + 7$