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

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Answer 1 · 2016-11-20T17:19:59

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

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

$f(x)=(x-a)(x-b)(x-c)(x-d)$

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

$f(x)=(x-1)(x-1)(x-i)(x-(-i))$

= $(x^2-2x+1)(x-i)(x+i)$

= $(x^2-2x+1)(x^2-i^2)$

= $(x^2-2x+1)(x^2+1)$ (as $i^2=-1$ )

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

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