document.write( "Question 694628: Factor this trinomial:\r
\n" ); document.write( "\n" ); document.write( "2t^2 - t - 3 \n" ); document.write( "

Algebra.Com's Answer #428087 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2t%5E2-t-3\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-3\"$ to get $\"%282%29%28-3%29=-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-6\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?\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 $\"-6\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-6\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6\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 $\"-6\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-6) = -6
\n" ); document.write( "2*(-3) = -6
\n" ); document.write( "(-1)*(6) = -6
\n" ); document.write( "(-2)*(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 $\"-1\"$ :\r
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First Number	Second Number	Sum
1	-6	1+(-6)=-5
2	-3	2+(-3)=-1
-1	6	-1+6=5
-2	3	-2+3=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-3\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-3\"$ both multiply to $\"-6\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1t\"$ with $\"2t-3t\"$ . Remember, $\"2\"$ and $\"-3\"$ add to $\"-1\"$ . So this shows us that $\"2t-3t=-1t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2t%5E2%2Bhighlight%282t-3t%29-3\"$ Replace the second term $\"-1t\"$ with $\"2t-3t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282t%5E2%2B2t%29%2B%28-3t-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2t%28t%2B1%29%2B%28-3t-3%29\"$ Factor out the GCF $\"2t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2t%28t%2B1%29-3%28t%2B1%29\"$ Factor out $\"3\"$ 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( " $\"%282t-3%29%28t%2B1%29\"$ Combine like terms. Or factor out the common term $\"t%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2t%5E2-t-3\"$ factors to $\"%282t-3%29%28t%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"2t%5E2-t-3=%282t-3%29%28t%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282t-3%29%28t%2B1%29\"$ to get $\"2t%5E2-t-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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