document.write( "Question 1206505: Use the Rational Root Theorem to list all possible rational solutions. Then find the actual rational solutions.\r
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\n" ); document.write( "\n" ); document.write( " 2x^4 - 5x^3 - 17x^2 + 41x - 21 = 0 \n" ); document.write( "
Algebra.Com's Answer #844036 by Edwin McCravy(20063)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "All candidates for rational solutions must have a numerator which\r\n" );
document.write( "divides evenly into the constant term in absolute value |-21|=21;\r\n" );
document.write( "and whose denominator divides evenly into the absolute value of \r\n" );
document.write( "the leading coefficient |2|=2.\r\n" );
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document.write( "candidates for numerators of rational solutions: 1,3,7,21\r\n" );
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document.write( "candidates for denominators of rational solutions: 1,2\r\n" );
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document.write( "candidates for rational solutions: \r\n" );
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document.write( "1/1, 1/2, 3/1, 3/2, 7/1, 7/2, 21/1, 21/2\r\n" );
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document.write( "or\r\n" );
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document.write( "1, 1/2, 3, 3/2, 7, 7/2, 21, 21/2\r\n" );
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document.write( "\"2x%5E4+-+5x%5E3+-+17x%5E2+%2B+41x+-+21+=+0\"\r\n" );
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document.write( "Try 1, using synthetic division:\r\n" );
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document.write( "1 | 2  -5 -17  41 -21\r\n" );
document.write( "  |     2  -3 -20  21 \r\n" );
document.write( "    2  -3 -20  21   0\r\n" );
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document.write( "That factors the left side as\r\n" );
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document.write( "\"%28x%5E%22%22-1%29%282x%5E3-3x%5E2-20x%2B21%29=0\"\r\n" );
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document.write( "The 0 on the bottom right tells us that 1 is a rational solution\r\n" );
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document.write( "Try 1 again in the quotient because it might have multiplicity \r\n" );
document.write( "more than 1.\r\n" );
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document.write( "1 | 2  -3 -20  21 \r\n" );
document.write( "  |     2  -1 -21  \r\n" );
document.write( "    2  -1 -21   0\r\n" );
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document.write( "That factors the left side again as\r\n" );
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document.write( "\"%28x%5E%22%22-1%29%28x%5E%22%22-1%29%282x%5E2-x-21%29=0\"\r\n" );
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document.write( "Again the 0 on the bottom right tells us that 1 is a second solution\r\n" );
document.write( "of at least multiplicity 2.\r\n" );
document.write( "or\r\n" );
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document.write( "\"%28x-1%5E%22%22%29%5E2%282x%5E2-x-21%29=0\"\r\n" );
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document.write( "We already know how to finish factoring, for it is a quadratic:\r\n" );
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document.write( "\"%28x-1%5E%22%22%29%5E2%282x-7%5E%22%22%29%28x%2B3%29=0\"\r\n" );
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document.write( "x-1=0;  2x-7=0;  x+3=0\r\n" );
document.write( "  x=1;    2x=7;    x=-3\r\n" );
document.write( "           x=7/2\r\n" );
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document.write( "So: \r\n" );
document.write( "1 is a rational solution with multiplicity 2.\r\n" );
document.write( "7/2 is a rational solution with multiplicity of multiplicity 1.\r\n" );
document.write( "-3 is a rational solution with multiplicity of multiplicity 1.\r\n" );
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document.write( "Edwin
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