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What all the zeros of the equation
x^(5) +x^(4) +x^(3) +x^(2) +x+1=0

User Rzelek
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1 Answer

5 votes

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.

<95141404393>

What all the zeros of the equation x^(5) +x^(4) +x^(3) +x^(2) +x+1=0-example-1
What all the zeros of the equation x^(5) +x^(4) +x^(3) +x^(2) +x+1=0-example-2
User LAS
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7.5k points

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