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

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Question 422795: Solve the polynomial equation.
x^6 - 19x^3 = 216
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Let z+=+x%5E3. Our polynomial becomes

z%5E2+-+19z+=+216
z%5E2+-+19z+-+216+=+0. We can factor this:
%28z-27%29%28z%2B8%29+=+0
z+=+27 or z+=+-8
Hence, all solutions x must satisfy x%5E3+=+27 or x%5E3+=+-8.
If you know roots of unity, we can conclude that the roots of x%5E3+=+27 are evenly spaced on the complex plane starting with x = 3. The other two roots are 3%28cos%282pi%2F3%29+%2B+i%2Asin%282pi%2F3%29%29 and 3%28cos%284pi%2F3%29+%2B+i%2Asin%284pi%2F3%29%29. For x%5E3+=+-8 we have -2, 2%28cos%285pi%2F3%29+%2B+i%2Asin%285pi%2F3%29%29 and 2%28cos%28pi%2F3%29+%2B+i%2Asin%28pi%2F3%29%29 as our roots for x.