document.write( "Question 552204: How to factor: x^2 +6x+40? \n" ); document.write( "

Algebra.Com's Answer #360181 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B6x%2B40\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"6\"$ , and the last term is $\"40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"40\"$ to get $\"%281%29%2840%29=40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"40\"$ (the previous product) and add to the second coefficient $\"6\"$ ?\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 $\"40\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"40\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,40\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40\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 $\"40\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*40 = 40
\n" ); document.write( "2*20 = 40
\n" ); document.write( "4*10 = 40
\n" ); document.write( "5*8 = 40
\n" ); document.write( "(-1)*(-40) = 40
\n" ); document.write( "(-2)*(-20) = 40
\n" ); document.write( "(-4)*(-10) = 40
\n" ); document.write( "(-5)*(-8) = 40\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 $\"6\"$ :\r
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First Number	Second Number	Sum
1	40	1+40=41
2	20	2+20=22
4	10	4+10=14
5	8	5+8=13
-1	-40	-1+(-40)=-41
-2	-20	-2+(-20)=-22
-4	-10	-4+(-10)=-14
-5	-8	-5+(-8)=-13

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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 $\"6\"$ . So $\"x%5E2%2B6x%2B40\"$ 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 $\"x%5E2%2B6x%2B40\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B6x%2B40\"$ is prime.
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