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受欢迎的代数 >

z^6+z^3+1=0

解答

z^{6} + z^{3} + 1 = 0

解答

z = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), z = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), z = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), z = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), z = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), z = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

求解步骤

z^{6} + z^{3} + 1 = 0

用

u = z^{3}

和

u^{2} = z^{6}

改写方程式

u^{2} + u + 1 = 0

解

u^{2} + u + 1 = 0 : u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

代回

u = z^{3}

，求解

z

解

z^{3} = - \frac{1}{2} + i \frac{3}{2} : z = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), z = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), z = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9})

解

z^{3} = - \frac{1}{2} - i \frac{3}{2} : z = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), z = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), z = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

解为

z = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), z = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), z = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), z = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), z = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), z = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

作图

流行的例子

x^{4} = \frac{16}{81}

x^3-7x^2+4x+12=0

x^{3} - 7 x^{2} + 4 x + 12 = 0

x^4-178x^2+3825=0

x^{4} - 178 x^{2} + 3825 = 0

-x^3+11x^2-38x+36=0

- x^{3} + 11 x^{2} - 38 x + 36 = 0

x^{3} = 24