# How do you write a polynomial in standard form given the zeros x=5, 8i, and -8i?

Nov 2, 2016

$f \left(x\right) = {x}^{3} - 5 {x}^{2} + 64 x - 320$

#### Explanation:

We find the simplest such polynomial by multiplying the corresponding linear factors:

$f \left(x\right) = \left(x - 5\right) \left(x - 8 i\right) \left(x + 8 i\right)$

$\textcolor{w h i t e}{f \left(x\right)} = \left(x - 5\right) \left({x}^{2} - {\left(8 i\right)}^{2}\right)$

$\textcolor{w h i t e}{f \left(x\right)} = \left(x - 5\right) \left({x}^{2} + 64\right)$

$\textcolor{w h i t e}{f \left(x\right)} = {x}^{3} - 5 {x}^{2} + 64 x - 320$

Any polynomial in $x$ with these zeros will be a multiple (scalar or polynomial) of this $f \left(x\right)$.