document.write( "Question 1158926: Find the number of positive integers n, \"1+%3C=n%3C=1000\", for which the polynomial \"x%5E2+%2B+x+-+n\" can be factored as the product of two linear factors with integer coefficients. \n" ); document.write( "
Algebra.Com's Answer #781936 by Edwin McCravy(20055)\"\" \"About 
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document.write( "The factorization of x² + x - n must be in the form\r\n" );
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document.write( "(x - p)(x + q) where p and q are positive integers.\r\n" );
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document.write( "We must have this identity for all values of x:\r\n" );
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document.write( " x² + x - n = (x - p)(x + q)\r\n" );
document.write( " x² + x - n = x² + qx - px - pq\r\n" );
document.write( " x² + x - n = x² + (q - p)x - pq\r\n" );
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document.write( "So q - p = 1\r\n" );
document.write( "      pq = n\r\n" );
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document.write( "       q = p + 1\r\n" );
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document.write( "     p(p+1) = n\r\n" );
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document.write( " \"1%3C=n%3C=1000\"\r\n" );
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document.write( "\"1%3C=p%28p%2B1%29%3C=1000\"\r\n" );
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document.write( "\"1%3C=p%5E2%2Bp%3C=1000\"\r\n" );
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document.write( "       \r\n" );
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document.write( "The only positive zero of the polynomial on the left is approximately 0.6,\r\n" );
document.write( "and the p²+p-1 is positive for and only for positive values of integer p greater\r\n" );
document.write( "than or equal to 1.\r\n" );
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document.write( "The only negative zero of the polynomial on the right is approximately 31.1,\r\n" );
document.write( "and the p²+p-1000 is negative only for positive values of p less than or equal\r\n" );
document.write( "to 31.  So p<=31\r\n" );
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document.write( "Therefore there are 31 ways the polynomial x² + x - n can be factored as the\r\n" );
document.write( "product of two linear factors with integer coefficients.\r\n" );
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document.write( "     factorization   when    and \r\n" );
document.write( "     (x-p)(x+q)        p=     n=\r\n" );
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document.write( " 1.  (x-1)(x+2)         1     2\r\n" );
document.write( " 2.  (x-2)(x+3)         2     6\r\n" );
document.write( " 3.  (x-3)(x+4)         3    12 \r\n" );
document.write( "…\r\n" );
document.write( "29.  (x-29)(x+30)      29   870\r\n" );
document.write( "30.  (x-30)(x+31)      30   930\r\n" );
document.write( "31.  (x-31)(x+32)      31   992\r\n" );
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document.write( "Edwin
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