document.write( "Question 319305: if α and β are zeroes of 3x^2+8x+2 ,find value of α^2+ß^2
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Algebra.Com's Answer #228619 by Edwin McCravy(20064)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The zeros \"alpha\" and \"beta\" of\r\n" );
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document.write( "\"3x%5E2%2B8x%2B2\" \r\n" );
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document.write( "are also the zeros of\r\n" );
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document.write( "\"x%5E2\"\"%22%2B%22\"\"8%2F3\"\"x\"\"%22%2B%22\"\"2%2F3\"\r\n" );
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document.write( "which is the original divided through by 3, the coefficient of \"x%5E2\".\r\n" );
document.write( "Known facts about the relationships between the coefficients and the \r\n" );
document.write( "zeros of a monic polynomial tell us that:\r\n" );
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document.write( "\"-%28alpha+%2B+beta%29+=+8%2F3\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"alpha%2Abeta+=+2%2F3\"\r\n" );
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document.write( "Since:\r\n" );
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document.write( "\"-%28alpha+%2B+beta%29+=+8%2F3\"\r\n" );
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document.write( "then multiplying both sides by -1:\r\n" );
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document.write( "\"alpha+%2B+beta+=+-8%2F3\"\r\n" );
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document.write( "Squaring both sides:\r\n" );
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document.write( "\"%28alpha+%2B+beta%29%5E2+=+%28-8%2F3%29%5E2\"\r\n" );
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document.write( "\"alpha%5E2+%2B+2alpha%2Abeta+%2B+beta%5E2+=+64%2F9\"\r\n" );
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document.write( "and now since \"alpha%2Abeta+=+2%2F3\", we substitute \"2%2F3\" for \"alpha%2Abeta\":\r\n" );
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document.write( "\"alpha%5E2+%2B+2%282%2F3%29+%2B+beta%5E2+=+64%2F9\"\r\n" );
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document.write( "\"alpha%5E2+%2B+4%2F3+%2B+beta%5E2+=+64%2F9\"\r\n" );
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document.write( "So \r\n" );
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document.write( "\"alpha%5E2+%2B+beta%5E2+=+64%2F9+-+4%2F3+=+64%2F9-12%2F9\"\r\n" );
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document.write( "\"alpha%5E2+%2B+beta%5E2+=+52%2F9\"\r\n" );
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document.write( "--------------------------\r\n" );
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document.write( "1) \"alpha%5E3%2Bbeta%5E3\"\r\n" );
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document.write( "We can find that by factoring the sum of two cubes.  We already\r\n" );
document.write( "have values for \"alpha%5E2%2Bbeta%5E2\" and \"alpha%2Abeta\".\r\n" );
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document.write( "----------------------------\r\n" );
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document.write( "2) \"alpha%5E4%2Bbeta%5E4\"\r\n" );
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document.write( "We have \r\n" );
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document.write( "\"alpha%5E2%2Bbeta%5E2=52%2F9\"\r\n" );
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document.write( "Squaring both sides gives:\r\n" );
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document.write( "\"alpha%5E4+%2B+2alpha%5E2beta%5E2+%2B+beta%5E4+=+2704%2F81\"\r\n" );
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document.write( "\"alpha%5E4+%2B+2%28alpha%2Abeta%29%5E2+%2B+beta%5E4+=+2704%2F81\"\r\n" );
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document.write( "And since \"alpha%2Abeta+=+2%2F3\",\r\n" );
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document.write( "\"alpha%5E4+%2B+2%282%2F3%29%5E2+%2B+beta%5E4+=+2704%2F81\"\r\n" );
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document.write( "\"alpha%5E4+%2B+2%284%2F9%29+%2B+beta%5E4+=+2704%2F81\"\r\n" );
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document.write( "\"alpha%5E4+%2B+8%2F9+%2B+beta%5E4+=+2704%2F81\"\r\n" );
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document.write( "\"alpha%5E4+%2B+beta%5E4+=+2704%2F81+-+8%2F9\"\r\n" );
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document.write( "\"alpha%5E4+%2B+beta%5E4+=+2704%2F81-+72%2F9\"\r\n" );
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document.write( "\"alpha%5E4+%2B+beta%5E4+=+2632%2F81\"\r\n" );
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document.write( "------------------\r\n" );
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document.write( "You stop there but we could show that \r\n" );
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document.write( "\"alpha%5E5+%2B+beta%5E5+=+-18848%2F243\"\r\n" );
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document.write( "and \r\n" );
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document.write( "\"alpha%5E6+%2B+beta%5E6+=+134992%2F729\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"alpha%5E7+%2B+beta%5E7+=+-966848%2F2187\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"alpha%5E8+%2B+beta%5E8+=+6924832%2F6561\"\r\n" );
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document.write( "and as far as we like.\r\n" );
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document.write( "Edwin
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