document.write( "Question 498680: x^6 - 511x^3-512 = 0\r
\n" ); document.write( "\n" ); document.write( "find all real solutions of the equation.\r
\n" ); document.write( "\n" ); document.write( "hi, im having trouble factoring this out..getting confused with the exponents.
\n" ); document.write( "thank you for your time! \n" ); document.write( "
Algebra.Com's Answer #337125 by Earlsdon(6294)\"\" \"About 
You can put this solution on YOUR website!
\"x%5E6-511x%5E3-512+=+0\" Treat this as a quadratic equation in \"x%5E3\" thus:
\n" ); document.write( "\"%28x%5E3%29%5E2-511%28x%5E3%29-512+=+0\" Now factor this.
\n" ); document.write( "\"%28x%5E3%2B1%29%28x%5E3-512%29+=+0\" Apply the \"zero product\" rule:
\n" ); document.write( "\"x%5E3%2B1+=+0\" or \"x%5E3-512+=+0\"
\n" ); document.write( "Now since the original equation is a 6th order equation, we can expect 6 roots.
\n" ); document.write( "Here's how we get them:
\n" ); document.write( "\"x%5E3%2B1+=+0\" This is the sum of cubes which can be factored thus:
\n" ); document.write( "\"x%5E3%2B1+=+%28x%2B1%29%28x%5E2-x%2B1%29\"so...
\n" ); document.write( "\"%28x%2B1%29%28x%5E2%2Bx%2B1%29+=+0\" Apply the \"zero product\" rule:
\n" ); document.write( "\"x%2B1+=+0\" so \"highlight%28x+=+-1%29\"
\n" ); document.write( "\"x%5E2%2Bx%2B1+=+0\" Apply the quadratic formula to get:
\n" ); document.write( "\"highlight%28x+=+5%2B0.866i%29\" and \"highlight%28x+=+5-0.866i%29\"
\n" ); document.write( "For the other three roots, we have:
\n" ); document.write( "\"x%5E3-512+=+0\" This is the difference of cubes \"%28x%29%5E3-%288%29%5E3\"which factors as:
\n" ); document.write( "\"%28x-8%29%28x%5E2%2B8x%2B64%29+=+0\" Apply the \"zero product\" rule:
\n" ); document.write( "\"x-8+=+0\" so \"highlight%28x+=+8%29\"
\n" ); document.write( "\"x%5E2%2B8x%2B64+=+0\" Apply the quadratic formula to get:
\n" ); document.write( "\"highlight%28x+=+-4%2B6.928i%29\" and \"highlight%28x+=+-4-6.928i%29\" \n" ); document.write( "
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