document.write( "Question 511918: Show 7(x^3)-6(x^2)+2x-1=0 has no integer solutions. \n" ); document.write( "

Algebra.Com's Answer #342454 by Edwin McCravy(20065) $\"\"$ $\"About$

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document.write( "7x³ - 6x² + 2x - 1 = 0\r\n" );
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document.write( "Every candidate for a rational solution is ± a fraction whose\r\n" );
document.write( "numerator is a divisor of the absolute value of the constant term,\r\n" );
document.write( "-1, and whose denominator is a divisor of the absolute value of the\r\n" );
document.write( "leading coefficient, 7.\r\n" );
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document.write( "The only divisor of |-1| is 1\r\n" );
document.write( "The only divisors of |7| are 1 and 7\r\n" );
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document.write( "Therefore the only candidates for rational solutions are\r\n" );
document.write( " and .  The only ones of those which\r\n" );
document.write( "are integers are 1 and -1, so we see if either of those is a \r\n" );
document.write( "solution:\r\n" );
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document.write( "We see if 1 is a solution using synthetic division:\r\n" );
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document.write( "1| 7  -6  2  -1\r\n" );
document.write( " |     7  1   3\r\n" );
document.write( "   7   1  3   2\r\n" );
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document.write( "That left a remainder of 2, not 0, so 1 is not a solution\r\n" );
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document.write( "We also see if -1 is a solution\r\n" );
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document.write( "-1| 7  -6  2  -1\r\n" );
document.write( "  |    -7 13 -15\r\n" );
document.write( "    7 -13 15 -16\r\n" );
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document.write( "That left a remainder of -16, not 0, so -1 is not a solution\r\n" );
document.write( "either.\r\n" );
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document.write( "Those two, 1 and -1,  were the only possible candidates for integer\r\n" );
document.write( "solutions.  So there are none.\r\n" );
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document.write( "Edwin

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