document.write( "Question 168715: factor, (show all stages, result of factoring out a GCF before taking resulting polynomial and factoring it again)Check answer: 24x^2-10x-4 \n" ); document.write( "

Algebra.Com's Answer #124337 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"24x%5E2-10x-4\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2812x%5E2-5x-2%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"12x%5E2-5x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12x%5E2-5x-2\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-2\"$ to get $\"%2812%29%28-2%29=-24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\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 $\"-24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"-24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-24)
\n" ); document.write( "2*(-12)
\n" ); document.write( "3*(-8)
\n" ); document.write( "4*(-6)
\n" ); document.write( "(-1)*(24)
\n" ); document.write( "(-2)*(12)
\n" ); document.write( "(-3)*(8)
\n" ); document.write( "(-4)*(6)\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 $\"-5\"$ :\r
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-8\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-8\"$ both multiply to $\"-24\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5x\"$ with $\"3x-8x\"$ . Remember, $\"3\"$ and $\"-8\"$ add to $\"-5\"$ . So this shows us that $\"3x-8x=-5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2%2Bhighlight%283x-8x%29-2\"$ Replace the second term $\"-5x\"$ with $\"3x-8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812x%5E2%2B3x%29%2B%28-8x-2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2B1%29%2B%28-8x-2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2B1%29-2%284x%2B1%29\"$ Factor out $\"2\"$ 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( " $\"%283x-2%29%284x%2B1%29\"$ Combine like terms. Or factor out the common term $\"4x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%2812x%5E2-5x-2%29\"$ and factors further to $\"2%283x-2%29%284x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"24x%5E2-10x-4\"$ completely factors to $\"2%283x-2%29%284x%2B1%29\"$ \r
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