# How do you write a polynomial function given the real zeroes 3-i,5i and coefficient 1?

Jan 10, 2016

$y = {x}^{4} - 6 {x}^{3} + 35 {x}^{2} - 150 x + 250$

#### Explanation:

Complex roots always come in pairs.

$3 - i$ will be a $0$, but so will $3 + i$.

$5 i$ will be a $0$, but so will $- 5 i$.

This gives us the function

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

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

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

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

$y = \left({x}^{2} - 6 x + 9 + 1\right) \left({x}^{2} + 25\right)$

$y = \left({x}^{2} - 6 x + 10\right) \left({x}^{2} + 25\right)$

$y = {x}^{4} - 6 {x}^{3} + 35 {x}^{2} - 150 x + 250$

Notice how the function only has imaginary roots (never crosses the $x$-axis):

graph{x^4-6x^3+35x^2-150x+250 [-5, 8, -36, 300]}