factorizar-x-4-1-

Question Number 204979 by Simurdiera last updated on 04/Mar/24

f a c t o r i z a r

x^{4} + 1

Answered by Skabetix last updated on 04/Mar/24

= (x - x_{0}) (x - x_{1}) (x - x_{2}) (x - x_{3})

= (x - e^{i \frac{Π}{4}}) (x - e^{\frac{3 i Π}{4}}) (x - e^{\frac{5 i Π}{4}}) (x - e^{\frac{7 i Π}{4}})

= (x^{2} - x \sqrt{} 2 + 1) (x^{2} + x \sqrt{} 2 + 1)

Answered by Frix last updated on 04/Mar/24

x^{4} + 1 =

= (x^{2} - \sqrt{2} x + 1) (x^{2} + \sqrt{2} x + 1) =

= (x - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i) (x + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i) (x - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i) (x + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i)

Answered by mr W last updated on 04/Mar/24

= x^{4} - i^{2}

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

= (x^{2} - {(\frac{- \sqrt{2} + \sqrt{2} i}{2})}^{2}) (x^{2} - {(\frac{\sqrt{2} + i \sqrt{2}}{2})}^{2})

= (x + \frac{- \sqrt{2} + \sqrt{2} i}{2}) (x - \frac{- \sqrt{2} + \sqrt{2} i}{2}) (x + \frac{\sqrt{2} + i \sqrt{2}}{2}) (x - \frac{\sqrt{2} + \sqrt{2} i}{2})

Answered by MATHEMATICSAM last updated on 05/Mar/24

x^{4} + 1

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

= {(x^{2} + 1)}^{2} - {(\sqrt{2} x)}^{2}

= (x^{2} + \sqrt{2} x + 1) (x^{2} - \sqrt{2} x + 1)