Factoring a complex polynomial

  • #1
TheCanadian
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I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or any particular method (possibly a general one) a person can use to find the factored form of the polynomial?
 

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  • #2
fresh_42
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I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or any particular method (possibly a general one) a person can use to find the factored form of the polynomial?
##1+z+z^2+\dots +z^{n-1} = \frac{z^n - 1}{z-1}##. I.e. the roots of ##z^n-1## are ##1## plus the roots of ##1+z+z^2+\dots +z^{n-1}##. They are called roots of unity (##z^n = 1 ##) and are equally distributed along the unit circle, starting at ##(1,0)##. Therefore they are multiples of the angle ##\frac{2 \pi}{n}##, i.e. ##\exp (i{\frac{k}{n} 2 \pi}) ## with ##k=0,\dots ,n-1##.
 
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