document.write( "Question 607717: What is the factored form of 12d squared + 4d - 1 ? \n" ); document.write( "

Algebra.Com's Answer #382864 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12d%5E2%2B4d-1\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"4\"$ , and the last term is $\"-1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-1\"$ to get $\"%2812%29%28-1%29=-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-12\"$ (the previous product) and add to the second coefficient $\"4\"$ ?\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 $\"-12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\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 $\"-12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-12) = -12
\n" ); document.write( "2*(-6) = -12
\n" ); document.write( "3*(-4) = -12
\n" ); document.write( "(-1)*(12) = -12
\n" ); document.write( "(-2)*(6) = -12
\n" ); document.write( "(-3)*(4) = -12\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 $\"4\"$ :\r
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First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"6\"$ add to $\"4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"6\"$ both multiply to $\"-12\"$ and add to $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"4d\"$ with $\"-2d%2B6d\"$ . Remember, $\"-2\"$ and $\"6\"$ add to $\"4\"$ . So this shows us that $\"-2d%2B6d=4d\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12d%5E2%2Bhighlight%28-2d%2B6d%29-1\"$ Replace the second term $\"4d\"$ with $\"-2d%2B6d\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812d%5E2-2d%29%2B%286d-1%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2d%286d-1%29%2B%286d-1%29\"$ Factor out the GCF $\"2d\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2d%286d-1%29%2B1%286d-1%29\"$ 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( " $\"%282d%2B1%29%286d-1%29\"$ Combine like terms. Or factor out the common term $\"6d-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12d%5E2%2B4d-1\"$ factors to $\"%282d%2B1%29%286d-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"12d%5E2%2B4d-1=%282d%2B1%29%286d-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282d%2B1%29%286d-1%29\"$ to get $\"12d%5E2%2B4d-1\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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