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 #781941 by solver91311(24713) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Since the sign on the constant term is negative, the two roots must have opposite signs. And since the coefficient on the linear term is a positive 1, the positive root must have an absolute value that exceeds the absolute value of the negative root by one.\r
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\r
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\n" ); document.write( "\n" ); document.write( "then there exist integers

and

such that if

and

are factors of the quadratic, the following relations hold:\r
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\r
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\n" ); document.write( "\n" ); document.write( "and\r
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\r
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\n" ); document.write( "\n" ); document.write( "We know the smallest possible values for

and

are 1 and 2, and therefore the smallest

is 2. To get a boundary for the largest possible

we solve:\r
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\n" ); document.write( "\n" ); document.write( "I leave it as an exercise for the student to verify that the positive root of this equation is slightly in excess of 31.\r
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\n" ); document.write( "\n" ); document.write( "Checking: If

, then

and

but if

, then =

and

\ 1000\">.\r
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\n" ); document.write( "\n" ); document.write( "Therefore the set of values for

that satisfies all conditions of the problem is

which leads to the conclusion that there are exactly 31

pairs that produce 31 unique values

that satisfy the conditions.
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\n" ); document.write( "John
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\n" ); document.write( "My calculator said it, I believe it, that settles it
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