document.write( "Question 179598: Factor the trinomial 42b^2-43b-7= \n" ); document.write( "

Algebra.Com's Answer #134518 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"42b%5E2-43b-7\"$ , we can see that the first coefficient is $\"42\"$ , the second coefficient is $\"-43\"$ , and the last term is $\"-7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"42\"$ by the last term $\"-7\"$ to get $\"%2842%29%28-7%29=-294\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-294\"$ (the previous product) and add to the second coefficient $\"-43\"$ ?\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 $\"-294\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-294\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,14,21,42,49,98,147,294\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42,-49,-98,-147,-294\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 $\"-294\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-294)
\n" ); document.write( "2*(-147)
\n" ); document.write( "3*(-98)
\n" ); document.write( "6*(-49)
\n" ); document.write( "7*(-42)
\n" ); document.write( "14*(-21)
\n" ); document.write( "(-1)*(294)
\n" ); document.write( "(-2)*(147)
\n" ); document.write( "(-3)*(98)
\n" ); document.write( "(-6)*(49)
\n" ); document.write( "(-7)*(42)
\n" ); document.write( "(-14)*(21)\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 $\"-43\"$ :\r
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First Number	Second Number	Sum
1	-294	1+(-294)=-293
2	-147	2+(-147)=-145
3	-98	3+(-98)=-95
6	-49	6+(-49)=-43
7	-42	7+(-42)=-35
14	-21	14+(-21)=-7
-1	294	-1+294=293
-2	147	-2+147=145
-3	98	-3+98=95
-6	49	-6+49=43
-7	42	-7+42=35
-14	21	-14+21=7

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"6\"$ and $\"-49\"$ add to $\"-43\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"-49\"$ both multiply to $\"-294\"$ and add to $\"-43\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-43b\"$ with $\"6b-49b\"$ . Remember, $\"6\"$ and $\"-49\"$ add to $\"-43\"$ . So this shows us that $\"6b-49b=-43b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"42b%5E2%2Bhighlight%286b-49b%29-7\"$ Replace the second term $\"-43b\"$ with $\"6b-49b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2842b%5E2%2B6b%29%2B%28-49b-7%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6b%287b%2B1%29%2B%28-49b-7%29\"$ Factor out the GCF $\"6b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6b%287b%2B1%29-7%287b%2B1%29\"$ Factor out $\"7\"$ 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( " $\"%286b-7%29%287b%2B1%29\"$ Combine like terms. Or factor out the common term $\"7b%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"42b%5E2-43b-7\"$ factors to $\"%286b-7%29%287b%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%286b-7%29%287b%2B1%29\"$ to get $\"42b%5E2-43b-7\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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