document.write( "Question 640515: Sorry if this is the wrong topic but
\n" ); document.write( "Is this even possible to factor out a binomial? \r
\n" ); document.write( "\n" ); document.write( "6x^2+17x-40 \n" ); document.write( "

Algebra.Com's Answer #403250 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2%2B17x-40\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"17\"$ , and the last term is $\"-40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-40\"$ to get $\"%286%29%28-40%29=-240\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-240\"$ (the previous product) and add to the second coefficient $\"17\"$ ?\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 $\"-240\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-240\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240\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 $\"-240\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-240) = -240
\n" ); document.write( "2*(-120) = -240
\n" ); document.write( "3*(-80) = -240
\n" ); document.write( "4*(-60) = -240
\n" ); document.write( "5*(-48) = -240
\n" ); document.write( "6*(-40) = -240
\n" ); document.write( "8*(-30) = -240
\n" ); document.write( "10*(-24) = -240
\n" ); document.write( "12*(-20) = -240
\n" ); document.write( "15*(-16) = -240
\n" ); document.write( "(-1)*(240) = -240
\n" ); document.write( "(-2)*(120) = -240
\n" ); document.write( "(-3)*(80) = -240
\n" ); document.write( "(-4)*(60) = -240
\n" ); document.write( "(-5)*(48) = -240
\n" ); document.write( "(-6)*(40) = -240
\n" ); document.write( "(-8)*(30) = -240
\n" ); document.write( "(-10)*(24) = -240
\n" ); document.write( "(-12)*(20) = -240
\n" ); document.write( "(-15)*(16) = -240\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 $\"17\"$ :\r
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First Number	Second Number	Sum
1	-240	1+(-240)=-239
2	-120	2+(-120)=-118
3	-80	3+(-80)=-77
4	-60	4+(-60)=-56
5	-48	5+(-48)=-43
6	-40	6+(-40)=-34
8	-30	8+(-30)=-22
10	-24	10+(-24)=-14
12	-20	12+(-20)=-8
15	-16	15+(-16)=-1
-1	240	-1+240=239
-2	120	-2+120=118
-3	80	-3+80=77
-4	60	-4+60=56
-5	48	-5+48=43
-6	40	-6+40=34
-8	30	-8+30=22
-10	24	-10+24=14
-12	20	-12+20=8
-15	16	-15+16=1

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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 $\"17\"$ . So $\"6x%5E2%2B17x-40\"$ 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 $\"6x%5E2%2B17x-40\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2%2B17x-40\"$ is prime.
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