document.write( "Question 615199: Factor by grouping. 12a^2-7a-5 \n" ); document.write( "

Algebra.Com's Answer #387007 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12a%5E2-7a-5\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"-7\"$ , and the last term is $\"-5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-5\"$ to get $\"%2812%29%28-5%29=-60\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-60\"$ (the previous product) and add to the second coefficient $\"-7\"$ ?\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 $\"-60\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-60\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60\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 $\"-60\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-60) = -60
\n" ); document.write( "2*(-30) = -60
\n" ); document.write( "3*(-20) = -60
\n" ); document.write( "4*(-15) = -60
\n" ); document.write( "5*(-12) = -60
\n" ); document.write( "6*(-10) = -60
\n" ); document.write( "(-1)*(60) = -60
\n" ); document.write( "(-2)*(30) = -60
\n" ); document.write( "(-3)*(20) = -60
\n" ); document.write( "(-4)*(15) = -60
\n" ); document.write( "(-5)*(12) = -60
\n" ); document.write( "(-6)*(10) = -60\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 $\"-7\"$ :\r
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First Number	Second Number	Sum
1	-60	1+(-60)=-59
2	-30	2+(-30)=-28
3	-20	3+(-20)=-17
4	-15	4+(-15)=-11
5	-12	5+(-12)=-7
6	-10	6+(-10)=-4
-1	60	-1+60=59
-2	30	-2+30=28
-3	20	-3+20=17
-4	15	-4+15=11
-5	12	-5+12=7
-6	10	-6+10=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"-12\"$ add to $\"-7\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"-12\"$ both multiply to $\"-60\"$ and add to $\"-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-7a\"$ with $\"5a-12a\"$ . Remember, $\"5\"$ and $\"-12\"$ add to $\"-7\"$ . So this shows us that $\"5a-12a=-7a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12a%5E2%2Bhighlight%285a-12a%29-5\"$ Replace the second term $\"-7a\"$ with $\"5a-12a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812a%5E2%2B5a%29%2B%28-12a-5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%2812a%2B5%29%2B%28-12a-5%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%2812a%2B5%29-1%2812a%2B5%29\"$ Factor out $\"1\"$ 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( " $\"%28a-1%29%2812a%2B5%29\"$ Combine like terms. Or factor out the common term $\"12a%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12a%5E2-7a-5\"$ factors to $\"%28a-1%29%2812a%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"12a%5E2-7a-5=%28a-1%29%2812a%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28a-1%29%2812a%2B5%29\"$ to get $\"12a%5E2-7a-5\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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