# How do you write the polynomial function with the least degree and zeroes i, 4?

The requested polynomial is

$\left(x - i\right) \left(x - 4\right) = {x}^{2} - x \left(4 + i\right) + 4 i = 0$

Sep 12, 2015

If the polynomial is also to have Real coefficients, then it must also have $- i$ as a zero, so:

$f \left(x\right) = \left(x - i\right) \left(x + i\right) \left(x - 4\right) = \left({x}^{2} + 1\right) \left(x - 4\right)$

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

#### Explanation:

Complex roots of polynomial equations with Real coefficients always occur in conjugate pairs.

Note that any polynomial that has these roots will be a multiple (scalar or polynomial) of $f$.