document.write( "Question 243968: factor by grouping 3x^2+19x-2x-6 \n" ); document.write( "

Algebra.Com's Answer #178647 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"3x%5E2%2B19x-2x-6\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2B17x-6\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2%2B17x-6\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"17\"$ , and the last term is $\"-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"-6\"$ to get $\"%283%29%28-6%29=-18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-18\"$ (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 $\"-18\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-18\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18\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 $\"-18\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-18)
\n" ); document.write( "2*(-9)
\n" ); document.write( "3*(-6)
\n" ); document.write( "(-1)*(18)
\n" ); document.write( "(-2)*(9)
\n" ); document.write( "(-3)*(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 $\"17\"$ :\r
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First Number	Second Number	Sum
1	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"18\"$ add to $\"17\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"18\"$ both multiply to $\"-18\"$ and add to $\"17\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"17x\"$ with $\"-x%2B18x\"$ . Remember, $\"-1\"$ and $\"18\"$ add to $\"17\"$ . So this shows us that $\"-x%2B18x=17x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-x%2B18x%29-6\"$ Replace the second term $\"17x\"$ with $\"-x%2B18x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-x%29%2B%2818x-6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-1%29%2B%2818x-6%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x-1%29%2B6%283x-1%29\"$ Factor out $\"6\"$ 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( " $\"%28x%2B6%29%283x-1%29\"$ Combine like terms. Or factor out the common term $\"3x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B19x-2x-6\"$ factors to $\"%28x%2B6%29%283x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2%2B19x-2x-6=%28x%2B6%29%283x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x%2B6%29%283x-1%29\"$ to get $\"3x%5E2%2B17x-6\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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