Answer:
- x = -1
- x = (1 +i√3)/2
- x = (1 -i√3)/2
- x = (-1 +i√3)/2
- x = (-1 -i√3)/2
Explanation:
You want to know the zeros of the equation x⁵ +x⁴ +x³ +x² +x +1 = 0.
Graph
We like to approach solving higher degree polynomials by graphing them. The attached graph shows the only real root is x = -1.
4th-degree factor
Using synthetic division to factor that out, we have ...
(x +1)(x⁴ +x² +1) = 0
The second factor can be written as ...
x⁴ +x² +1 = (x⁴ +2x² +1) -x² = (x² +1)² -x²
Quadratic factors
As a "difference of squares", this can be further factored to the quadratic factors ...
(x² +1)² -x² = (x² -x +1)(x² +x +1)
Each of these factors can be rewritten by completing the squares:
= ((x² -x +1/4) +3/4)((x² -x +1/4) +3/4)
= ((x -1/2)² +3/4)((x +1/2)² +3/4)
And those can be factored using the "difference of squares" factoring.
= (x -1/2 -√(3/4)i)(x -1/2 +√(3/4)i)·(x +1/2 -√(3/4)i)(x +1/2 +√(3/4)i)
Complex roots
These factors tell us the four complex roots are ...
- x = ±1/2 ±i(√3)/2 . . . . . . . for all combinations of signs
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Additional comment
The second attachment shows the result from a calculator's "solver" function.
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