document.write( "Question 582096: Does anyone have a easy way to factor? Her is a problem I just don't know how to do.\r
\n" ); document.write( "\n" ); document.write( "21x^2-25x-4\r
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Algebra.Com's Answer #371931 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"21x%5E2-25x-4\"$ , we can see that the first coefficient is $\"21\"$ , the second coefficient is $\"-25\"$ , and the last term is $\"-4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"21\"$ by the last term $\"-4\"$ to get $\"%2821%29%28-4%29=-84\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-84\"$ (the previous product) and add to the second coefficient $\"-25\"$ ?\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 $\"-84\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-84\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,12,14,21,28,42,84\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84\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 $\"-84\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-84) = -84
\n" ); document.write( "2*(-42) = -84
\n" ); document.write( "3*(-28) = -84
\n" ); document.write( "4*(-21) = -84
\n" ); document.write( "6*(-14) = -84
\n" ); document.write( "7*(-12) = -84
\n" ); document.write( "(-1)*(84) = -84
\n" ); document.write( "(-2)*(42) = -84
\n" ); document.write( "(-3)*(28) = -84
\n" ); document.write( "(-4)*(21) = -84
\n" ); document.write( "(-6)*(14) = -84
\n" ); document.write( "(-7)*(12) = -84\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 $\"-25\"$ :\r
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First Number	Second Number	Sum
1	-84	1+(-84)=-83
2	-42	2+(-42)=-40
3	-28	3+(-28)=-25
4	-21	4+(-21)=-17
6	-14	6+(-14)=-8
7	-12	7+(-12)=-5
-1	84	-1+84=83
-2	42	-2+42=40
-3	28	-3+28=25
-4	21	-4+21=17
-6	14	-6+14=8
-7	12	-7+12=5

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-28\"$ add to $\"-25\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-28\"$ both multiply to $\"-84\"$ and add to $\"-25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-25x\"$ with $\"3x-28x\"$ . Remember, $\"3\"$ and $\"-28\"$ add to $\"-25\"$ . So this shows us that $\"3x-28x=-25x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"21x%5E2%2Bhighlight%283x-28x%29-4\"$ Replace the second term $\"-25x\"$ with $\"3x-28x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2821x%5E2%2B3x%29%2B%28-28x-4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%287x%2B1%29%2B%28-28x-4%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%287x%2B1%29-4%287x%2B1%29\"$ Factor out $\"4\"$ 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( " $\"%283x-4%29%287x%2B1%29\"$ Combine like terms. Or factor out the common term $\"7x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"21x%5E2-25x-4\"$ factors to $\"%283x-4%29%287x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"21x%5E2-25x-4=%283x-4%29%287x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x-4%29%287x%2B1%29\"$ to get $\"21x%5E2-25x-4\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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