document.write( "Question 588384: The factors of 2x^2+5x+7 \n" ); document.write( "

Algebra.Com's Answer #374480 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B5x%2B7\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"5\"$ , and the last term is $\"7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"7\"$ to get $\"%282%29%287%29=14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"14\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"14\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"14\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"14\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*14 = 14
\n" ); document.write( "2*7 = 14
\n" ); document.write( "(-1)*(-14) = 14
\n" ); document.write( "(-2)*(-7) = 14\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"5\"$ :\r
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First Number	Second Number	Sum
1	14	1+14=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"5\"$ . So $\"2x%5E2%2B5x%2B7\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2%2B5x%2B7\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2%2B5x%2B7\"$ is prime.\r
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