1-solve-z-4-1-i-3-2-factorize-p-x-x-4-1-i-3-inside-C-x-3-factorze-inside-R-x-the-polynom-p-x-

Question Number 49248 by maxmathsup by imad last updated on 04/Dec/18

1) s o l v e z^{4} = 1 + i \sqrt{3}

2) f a c t o r i z e p (x) = x^{4} - 1 - i \sqrt{3} i n s i d e C [x]

3) f a c t o r z e i n s i d e R [x] t h e p o l y n o m p (x) .

Answered by Smail last updated on 05/Dec/18

z^{4} = 2 e^{i (\frac{π}{3} + 2 k π)}

z = \sqrt[4]{2} e^{i (\frac{π}{12} + \frac{k π}{2})}

z_{0} = \sqrt[4]{2} e^{i \frac{π}{12}}, z_{1} = i \sqrt[4]{2} e^{i \frac{π}{12}} = {i z}_{0}

z_{2} = - \sqrt[4]{2} e^{i \frac{π}{12}} = - z_{0}, z_{3} = - {i z}_{0}

2)

p (x) = x^{4} - z^{4} = (x - z_{0}) (x + z_{0}) (x - {i z}_{0}) (x + {i z}_{0})

3)

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