SOLUTION: Solve the polynomial equation. x^6

SOLUTION: Solve the polynomial equation. x^6 - 19x^3 = 216

Question 422795: Solve the polynomial equation.
x^6 - 19x^3 = 216

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Let . Our polynomial becomes

. We can factor this:

or
Hence, all solutions x must satisfy or .
If you know roots of unity, we can conclude that the roots of are evenly spaced on the complex plane starting with x = 3. The other two roots are and . For we have , and as our roots for x.
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