document.write( "Question 1142740: The equation \"x%5E4-5x%5E3-9x%5E2%2Bax%2Bb=0\" has a triple root. Given this root is
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Algebra.Com's Answer #763497 by Edwin McCravy(20066)\"\" \"About 
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The equation \"x%5E4-5x%5E3-9x%5E2%2Bax%2Bb=0\" has a triple root. Given this root
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document.write( "It is well-known that for the 4th degree polynomial:\r\n" );
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document.write( "\"Ax%5E4+%2B+Bx%5E3+%2B+Cx%5E2+%2B+Dx+%2B+E+=+0\"\r\n" );
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document.write( "The sum of its roots taken one at a time = -B/A\r\n" );
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document.write( "The sum of all products of its roots taken two at a time = C/A\r\n" );
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document.write( "The sum of all products of its roots taken three at a time = -D/A\r\n" );
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document.write( "The sum of all products of its roots taken four at a time = E/A\r\n" );
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document.write( "In \"x%5E4-5x%5E3-9x%5E2%2Bax%2Bb=0\" we are told that one root is a\r\n" );
document.write( "triple root.  \r\n" );
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document.write( "So let the four roots be r, r, r, and s\r\n" );
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document.write( "Then A=1, B=-5, C=-9, D=a, E=b\r\n" );
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document.write( "The sum of roots taken one at a time = \r\n" );
document.write( "r+r+r+s = -(-5)/1 or 3r+s = 5\r\n" );
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document.write( "The sum of all products of its roots taken two at a time = \r\n" );
document.write( "rr+rr+rs+rr+rs+rs = -9/1 or 3r²+3rs = -9 or r²+rs = -3\r\n" );
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document.write( "So we have this system of equations:\r\n" );
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document.write( "\"system%283r%2Bs=5%2Cr%5E2%2Brs=-3%29\" \r\n" );
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document.write( "Solve the first equation for s, s = 5-3r\r\n" );
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document.write( "Substituting in the second equation,\r\n" );
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document.write( "\"r%5E2%2Br%285-3r%29=-3%29\"\r\n" );
document.write( "\"r%5E2%2B5r-3r%5E2=-3%29\"\r\n" );
document.write( "\"-2r%5E2%2B5r%2B3=0\"\r\n" );
document.write( "\"2r%5E2-5r-3=0\"\r\n" );
document.write( "\"%28r-3%29%282r%2B1%29=0\"\r\n" );
document.write( "r-3 = 0;  2r+1 = 0\r\n" );
document.write( "  r = 3     2r = -1\r\n" );
document.write( "             r = -1/2\r\n" );
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document.write( "We are told that the triple root is an integer, so we\r\n" );
document.write( "discard the -1/2, and take r = 3\r\n" );
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document.write( "Substituting r = 3 in \r\n" );
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document.write( "s = 5-3r\r\n" );
document.write( "s = 5-3(3)\r\n" );
document.write( "s = 5-9\r\n" );
document.write( "s = -4\r\n" );
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document.write( "The sum of all products of its roots taken three at a time = -a/1, so\r\n" );
document.write( "rrr+rrs+rrs+rrs = -a or r³+3r²s = -a, substituting for r and s:\r\n" );
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document.write( "     r³+3r²s = -a \r\n" );
document.write( "3³+3(3)²(-4) = -a\r\n" );
document.write( " 27+3(9)(-4) = -a\r\n" );
document.write( "      27-109 = -a\r\n" );
document.write( "         -81 = -a\r\n" );
document.write( "           a = 81\r\n" );
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document.write( "The sum of all products of its roots taken four at a time = b/1\r\n" );
document.write( "That's all of its roots, and there is but one product, so\r\n" );
document.write( "rrrs = b, or\r\n" );
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document.write( "   r³s = b\r\n" );
document.write( "3³(-4) = b\r\n" );
document.write( "27(-4) = b\r\n" );
document.write( "  -108 = b\r\n" );
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document.write( "Edwin
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