# How do you form a polynomial function whose zeros are 3 and 4 + i?

Feb 14, 2016

Convert zeros to factors and multiply out to find the simplest possible polynomials.

#### Explanation:

If $a$ is a zero of a polynomial $f \left(x\right)$, then $\left(x - a\right)$ is a factor and vice versa.

So the simplest polynomial with these zeros is:

$\left(x - 3\right) \left(x - \left(4 + i\right)\right) = {x}^{2} - \left(7 + i\right) x + \left(12 + 3 i\right)$

If you want Real coefficients, then the Complex conjugate $4 - i$ must also be a zero and we find the simplest polynomial is:

$\left(x - 3\right) \left(x - \left(4 + i\right)\right) \left(x - \left(4 - i\right)\right)$

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

$= \left(x - 3\right) \left({x}^{2} - 8 x + 17\right)$

$= {x}^{3} - 11 {x}^{2} + 41 x - 51$

Any polynomial with these zeros must be a multiple (scalar or polynomial) of these 'simplest' polynomials.