# How do you write a polynomial function of least degree with integral coefficients that has the given zeroes 1 + 2i, 1- i?

Mar 11, 2016

${x}^{4} - 4 {x}^{3} + 11 {x}^{2} - 14 x + 10$

#### Explanation:

The first notion is that a polynomial with zeroes that have imaginary components must have pairs of conjugate complex roots so that the polynomial has real coefficients

Then the polynomial of the problem must have these roots
$1 + 2 i , 1 - 2 i , 1 - i , 1 + 1$

So the required polynomial is

$\left(x - 1 + 2 i\right) \left(x - 1 - 2 i\right) \left(x - 1 + i\right) \left(x - 1 - i\right)$
$\left({x}^{2} - 2 x + 1 + 4\right) \left({x}^{2} - 2 x + 1 + 1\right)$
$\left({x}^{2} - 2 x + 5\right) \left({x}^{2} - 2 x + 2\right)$
${x}^{4} - 2 {x}^{3} + 2 {x}^{2} - 2 {x}^{3} + 4 {x}^{2} - 4 x + 5 {x}^{2} - 10 x + 10$
$\to {x}^{4} - 4 {x}^{3} + 11 {x}^{2} - 14 x + 10$