# How do you find a polynomial function of lowest degree with rational coefficients that has the given number of some of it's zeros. -5i, 3?

Jul 27, 2015

$f \left(x\right) = \left(x - 5 i\right) \left(x + 5 i\right) \left(x - 3\right) = {x}^{3} - 3 {x}^{2} + 25 x - 75$

#### Explanation:

If the coefficients are real (let alone rational), then any complex zeros will occur in conjugate pairs.

So the roots of $f \left(x\right) = 0$ are at least $\pm 5 i$ and $3$.

Hence

$f \left(x\right) = \left(x - 5 i\right) \left(x + 5 i\right) \left(x - 3\right)$

$= \left({x}^{2} + 25\right) \left(x - 3\right) = {x}^{3} - 3 {x}^{2} + 25 x - 75$

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