x^4+x^3+ X^2+x +1= 0

Question

solve x^4+x^3+x^2+x^2+x+1=0 find the complex roots.

Answer 1

(x-1)(x⁴+x³+x²+x+1)=x⁵-1.

When x5-1=0, x5=1, and the real solution to this is x=1. The other zeroes are complex and can be represented geometrically by a regular pentagon in an Argand diagram. In this diagram, the x value is the real component and the y value the imaginary component. Using trig we can identify all the complex zeroes. The pentagon is made up of 5 isosceles triangles with an apex angle of 360/5=72°. The real zero is represented by the point (1,0) on a graph. The first complex zero is represented by a radius of 1 at an angle of 72° to the x-axis. This gives us the point (cos72,sin72)=(0.3090,0.9511) which corresponds to complex number 0.3090+0.9511i.

The next zero is at (cos144,sin144)=(-cos36,sin36)=-0.8090+0.5878i.

The next zero is at (cos216,sin216)=(-cos36,-sin36)=-0.8090-0.5878i, then (cos288,sin288)=(cos72,-sin72)=0.3090-0.9511i.

This gives us two conjugate pairs: 

-0.8090+0.5878i and -0.8090-0.5878i, and 0.3090+0.9511i and 0.3090-0.9511i.

These are the complex zeroes.