# How do you write a polynomial with zeros 4-i and sqrt(10)?

Aug 16, 2016

This one works:

${x}^{2} - \left(4 + \sqrt{10} - i\right) x + \left(4 \sqrt{10} - \left(\sqrt{10}\right) i\right)$

...but you probably want this one:

${x}^{4} - 8 {x}^{3} + 7 {x}^{2} + 80 x - 170$

#### Explanation:

If you allow coefficients of arbitrary type, then the polynomial of lowest degree with these zeros is simply:

$\left(x - \left(4 - i\right)\right) \left(x - \sqrt{10}\right)$

$= {x}^{2} - \left(4 + \sqrt{10} - i\right) x + \left(4 \sqrt{10} - \left(\sqrt{10}\right) i\right)$

If you want rational coefficients, then include the Complex conjugate $4 + i$ and the radical conjugate $- \sqrt{10}$ as two more zeros to find:

$\left(x - \left(4 - i\right)\right) \left(x - \left(4 + i\right)\right) \left(x - \sqrt{10}\right) \left(x + \sqrt{10}\right)$

$= \left({\left(x - 4\right)}^{2} - {i}^{2}\right) \left({x}^{2} - 10\right)$

$= \left({x}^{2} - 8 x + 17\right) \left({x}^{2} - 10\right)$

$= {x}^{4} - 8 {x}^{3} + 7 {x}^{2} + 80 x - 170$