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

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

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

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

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

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

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

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

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

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