How do you write a polynomial function with minimum degree whose zeroes are 1-2i and 1+2i?

Jan 5, 2017

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

Explanation:

The requested polynomial function is:

$f \left(x\right) = \left(x - \left(1 - 2 i\right)\right) \left(x - \left(1 + 2 i\right)\right)$

that's:

$\left(x - 1 + 2 i\right) \left(x - 1 - 2 i\right)$

Since

$\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$

you get

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

Since

${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

and

${i}^{2} = - 1$

you get

${x}^{2} - 2 x + 1 + 4$

${x}^{2} - 2 x + 5$