document.write( "Question 626497: Factor x^2 - 19x - 120 \n" ); document.write( "

Algebra.Com's Answer #394221 by jim_thompson5910(35256) $\"\"$ $\"About$

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Looking at the expression $\"x%5E2-19x-120\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-19\"$ , and the last term is $\"-120\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-120\"$ to get $\"%281%29%28-120%29=-120\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-120\"$ (the previous product) and add to the second coefficient $\"-19\"$ ?\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 $\"-120\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-120\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120\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 $\"-120\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-120) = -120
\n" ); document.write( "2*(-60) = -120
\n" ); document.write( "3*(-40) = -120
\n" ); document.write( "4*(-30) = -120
\n" ); document.write( "5*(-24) = -120
\n" ); document.write( "6*(-20) = -120
\n" ); document.write( "8*(-15) = -120
\n" ); document.write( "10*(-12) = -120
\n" ); document.write( "(-1)*(120) = -120
\n" ); document.write( "(-2)*(60) = -120
\n" ); document.write( "(-3)*(40) = -120
\n" ); document.write( "(-4)*(30) = -120
\n" ); document.write( "(-5)*(24) = -120
\n" ); document.write( "(-6)*(20) = -120
\n" ); document.write( "(-8)*(15) = -120
\n" ); document.write( "(-10)*(12) = -120\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 $\"-19\"$ :\r
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First Number	Second Number	Sum
1	-120	1+(-120)=-119
2	-60	2+(-60)=-58
3	-40	3+(-40)=-37
4	-30	4+(-30)=-26
5	-24	5+(-24)=-19
6	-20	6+(-20)=-14
8	-15	8+(-15)=-7
10	-12	10+(-12)=-2
-1	120	-1+120=119
-2	60	-2+60=58
-3	40	-3+40=37
-4	30	-4+30=26
-5	24	-5+24=19
-6	20	-6+20=14
-8	15	-8+15=7
-10	12	-10+12=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"-24\"$ add to $\"-19\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"-24\"$ both multiply to $\"-120\"$ and add to $\"-19\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-19x\"$ with $\"5x-24x\"$ . Remember, $\"5\"$ and $\"-24\"$ add to $\"-19\"$ . So this shows us that $\"5x-24x=-19x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%285x-24x%29-120\"$ Replace the second term $\"-19x\"$ with $\"5x-24x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B5x%29%2B%28-24x-120%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B5%29%2B%28-24x-120%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B5%29-24%28x%2B5%29\"$ Factor out $\"24\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-24%29%28x%2B5%29\"$ Combine like terms. Or factor out the common term $\"x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-19x-120\"$ factors to $\"%28x-24%29%28x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2-19x-120=%28x-24%29%28x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-24%29%28x%2B5%29\"$ to get $\"x%5E2-19x-120\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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