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1.求方程式的根0=x^4+x^2+1

=(y-1)(y^2+y+1).....y=x^2

=y^3-1

y^3=1=cos360°+jsin360°

y=(cos360°+jsin360°)^(1/3)

y1=cos120°+jsin120°.....戴美弗定理

y2=cos240°+jsin240°

y3=cos360°+jsin360°

x1=+-(cos120°+jsin120°)^0.5

=+-(cos60°+jsin60°)

=+-(1+j√3)/2

x2=+-(cos240°+jsin240°)^0.5

=+-(cos120°+jsin120°)

=+-(-1+j√3)/2

2.求x^5+x^4+x^3+x^2+x+1=0所圍成五邊形之面積

0=x^6-1=(x-1)(x^5+x^4+x^3+x^2+x+1)

x^6=1=cos360°+jsin360°

x=(cos360°+jsin360°)^(1/6)

x1=cos60°+jsin60°.....戴美弗定理

x2=cos120°+jsin120°

x3=cos180°+jsin180°

x4=cos240°+jsin240°

x5=cos300°+jsin300°

此5邊形: 每兩者夾角=60°,每邊長=1

只有x1,x5的夾角=120°

所以面積為:

A=4pc*0.5*1*1*sin60°+0.5*1*1*sin120°

=2*√3/2+0.5*√3/2

=√3+√3/4

=5√3/4.....ans

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