document.write( "Question 227692: 2x2 + bx – 15 Find all positive and negative integers b for which the polynomial can be factored. \n" ); document.write( "

Algebra.Com's Answer #169148 by jsmallt9(3758) $\"\"$ $\"About$

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An expression like $\"2x%5E2%2Bbx-15\"$ , if it factors at all, will factor into the form: $\"%28a%5B1%5D%2Ax+%2B+c%5B1%5D%29%28a%5B2%5D%2Ax+%2B+c%5B2%5D%29\"$ where $\"a%5B1%5D\"$ and $\"a%5B2%5D\"$ are factors of the 2 and $\"c%5B1%5D\"$ and $\"c%5B2%5D\"$ are factors of the -15. And the $\"b+=+a%5B1%5D+%2A+c%5B2%5D+%2B+a%5B2%5D%2Ac%5B1%5D\"$

\n" ); document.write( "So to find all the possible b's we need to find the factors of 2 and -15:
\n" ); document.write( "Factors of 2:
\n" ); document.write( " 1 and 2
\n" ); document.write( "Factors of -15:
\n" ); document.write( " -1 and 15
\n" ); document.write( " 1 and -15
\n" ); document.write( " 3 and -5
\n" ); document.write( " -3 and 5
\n" ); document.write( "Now we just use all the possible combinations of factors of 2 and factors of -15 in $\"b+=+a%5B1%5D+%2A+c%5B2%5D+%2B+a%5B2%5D%2Ac%5B1%5D\"$ :

$\"b+=+1%2A%28-1%29+%2B+2%2A15+=+-1+%2B+30+=+29\"$
$\"b+=+1%2A%2815%29+%2B+2%2A%28-1%29+=+15+%2B+-2+=+13\"$
$\"b+=+1%2A1+%2B+2%2A%28-15%29+=+1+%2B+%28-30%29+=+-29\"$
$\"b+=+1%2A%28-15%29+%2B+2%2A1+=+-15+%2B+2+=+-13\"$
$\"b+=+1%2A%283%29+%2B+2%2A%28-5%29+=+3+%2B+%28-10%29+=+-7\"$
$\"b+=+1%2A%28-5%29+%2B+2%2A3+=+-5+%2B+6+=+1\"$
$\"b+=+1%2A%28-3%29+%2B+2%2A5+=+-3+%2B+10+=+7\"$
$\"b+=+1%2A5+%2B+2%2A%28-3%29+=+5+%2B+%28-6%29+=+-1\"$

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