# How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 1, 1, i, -i?

Nov 20, 2016

Function is ${x}^{4} - 2 {x}^{3} + 2 {x}^{2} - 2 x + 1$

#### Explanation:

A function with leading coefficient as $1$ and zeros as $a$, $b$, $c$ and $d$ is

$f \left(x\right) = \left(x - a\right) \left(x - b\right) \left(x - c\right) \left(x - d\right)$

Hence if zeros are $1$, $1$, $i$ and $- i$, the function is

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

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

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

= $\left({x}^{2} - 2 x + 1\right) \left({x}^{2} + 1\right)$ (as ${i}^{2} = - 1$)

= ${x}^{4} - 2 {x}^{3} + {x}^{2} + {x}^{2} - 2 x + 1$

= ${x}^{4} - 2 {x}^{3} + 2 {x}^{2} - 2 x + 1$