Liczby zespolone

[wsl1993_]

Rozłożyć na czynniki (o współczynnikach rzeczywistych i zespolonych).
\(a)z^4-1
b)z^2+z+1
c)z^4+1
d)z^4-2z^2-3
e)jz^2-z+2j\)

[josselyn]

\(a)
z^4-1=(z^2-1)(z^2+1)=(z-1)(z+1)(z-i)(z+i)\)

[irena]

b)
\(z^2+z+1=(z+\frac{1+\sqrt{3}i}{2})(z+\frac{1-\sqrt{3}i}{2})\)

\(\Delta=1-4=-3\\\sqrt{\Delta}=\sqrt{3}i\\z_1=\frac{-1-\sqrt{3}+i}{2}\ \vee\ z_2=\frac{-1+\sqrt{3}i}{2}\)

[irena]

c)
\(z^4+1=(z^2-i)(z^2+i)=(z+\sqrt{i})(z-\sqrt{i})(z-i\sqrt{i})(z+i\sqrt{i})\)

\(z^2+1=0\\\Delta=-4i\\\sqrt{\Delta}=2i\sqrt{i}\\z_1=\frac{-2i\sqrt{i}}{2}=-i\sqrt{i}\ \vee\ z_2=\frac{2i\sqrt{i}}{2}=i\sqrt{i}\)

[irena]

d)
\(z^4-2z^2-3=(z^2+1)(z^2-3)=(z-i)(z+i)(z-\sqrt{3})(z+\sqrt{3})\)

[irena]

e)
\(iz^2-z+2i=i(z-i)(z+2i)=(iz-i^2)(z+2i)=(iz+1)(z+2i)\)

\(\Delta=1-8i^2=1+8=9\\z_1=\frac{1-3}{2i}=\frac{-2}{2i}=\frac{-i}{i^2}=\frac{-i}{-1}=i\ \vee\ z_2=\frac{1+3}{2i}=\frac{4}{2i}=\frac{4i}{-2}=-2i\)