document.write( "Question 548530: I need help factoring the trinomial t^2-24t+135
\n" ); document.write( "I got the answer (t-27) (t-3) but when you use the FOIL method to check you get t^2-30t+135. I think that all I have to do is switch some signs but I don't know how to do that. \n" ); document.write( "

Algebra.Com's Answer #357043 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 $\"t%5E2-24t%2B135\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-24\"$ , and the last term is $\"135\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"135\"$ to get $\"%281%29%28135%29=135\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"135\"$ (the previous product) and add to the second coefficient $\"-24\"$ ?\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 $\"135\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"135\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,9,15,27,45,135\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-9,-15,-27,-45,-135\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 $\"135\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*135 = 135
\n" ); document.write( "3*45 = 135
\n" ); document.write( "5*27 = 135
\n" ); document.write( "9*15 = 135
\n" ); document.write( "(-1)*(-135) = 135
\n" ); document.write( "(-3)*(-45) = 135
\n" ); document.write( "(-5)*(-27) = 135
\n" ); document.write( "(-9)*(-15) = 135\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 $\"-24\"$ :\r
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First Number	Second Number	Sum
1	135	1+135=136
3	45	3+45=48
5	27	5+27=32
9	15	9+15=24
-1	-135	-1+(-135)=-136
-3	-45	-3+(-45)=-48
-5	-27	-5+(-27)=-32
-9	-15	-9+(-15)=-24

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-9\"$ and $\"-15\"$ add to $\"-24\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-9\"$ and $\"-15\"$ both multiply to $\"135\"$ and add to $\"-24\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-24t\"$ with $\"-9t-15t\"$ . Remember, $\"-9\"$ and $\"-15\"$ add to $\"-24\"$ . So this shows us that $\"-9t-15t=-24t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"t%5E2%2Bhighlight%28-9t-15t%29%2B135\"$ Replace the second term $\"-24t\"$ with $\"-9t-15t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28t%5E2-9t%29%2B%28-15t%2B135%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t-9%29%2B%28-15t%2B135%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t-9%29-15%28t-9%29\"$ Factor out $\"15\"$ 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( " $\"%28t-15%29%28t-9%29\"$ Combine like terms. Or factor out the common term $\"t-9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"t%5E2-24t%2B135\"$ factors to $\"%28t-15%29%28t-9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"t%5E2-24t%2B135=%28t-15%29%28t-9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28t-15%29%28t-9%29\"$ to get $\"t%5E2-24t%2B135\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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