# How do you write a polynomial function of least degree with integral coefficients that has the given zeroes 6, 2i?

Jul 17, 2016

$f \left(x\right) = {x}^{3} - 6 {x}^{2} + 4 x - 24$

#### Explanation:

If a polynomial has Real coefficients then any non-Real zeros will occur in Complex conjugate pairs. So if $2 i$ is a zero, then so is $- 2 i$.

Hence our polynomial function can be written:

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

$= \left(x - 6\right) \left({x}^{2} + 4\right)$

$= {x}^{3} - 6 {x}^{2} + 4 x - 24$

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