document.write( "Question 422795: Solve the polynomial equation.
\n" ); document.write( "x^6 - 19x^3 = 216
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Algebra.Com's Answer #295182 by richard1234(7193) $\"\"$ $\"About$

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Let $\"z+=+x%5E3\"$ . Our polynomial becomes\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2+-+19z+=+216\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z%5E2+-+19z+-+216+=+0\"$ . We can factor this:\r
\n" ); document.write( "\n" ); document.write( " $\"%28z-27%29%28z%2B8%29+=+0\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z+=+27\"$ or $\"z+=+-8\"$ \r
\n" ); document.write( "\n" ); document.write( "Hence, all solutions x must satisfy $\"x%5E3+=+27\"$ or $\"x%5E3+=+-8\"$ .\r
\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "

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