document.write( "Question 697929: how to factor 4x^2-2x+7 \n" ); document.write( "

Algebra.Com's Answer #430346 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4x%5E2-2x%2B7\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"7\"$ to get $\"%284%29%287%29=28\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"28\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\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 $\"28\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"28\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,14,28\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-14,-28\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 $\"28\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*28 = 28
\n" ); document.write( "2*14 = 28
\n" ); document.write( "4*7 = 28
\n" ); document.write( "(-1)*(-28) = 28
\n" ); document.write( "(-2)*(-14) = 28
\n" ); document.write( "(-4)*(-7) = 28\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 $\"-2\"$ :\r
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First Number	Second Number	Sum
1	28	1+28=29
2	14	2+14=16
4	7	4+7=11
-1	-28	-1+(-28)=-29
-2	-14	-2+(-14)=-16
-4	-7	-4+(-7)=-11

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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 $\"-2\"$ . So $\"4x%5E2-2x%2B7\"$ 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 $\"4x%5E2-2x%2B7\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-2x%2B7\"$ is prime.
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