document.write( "Question 293740: Factor: tsquared+3t-10 \n" ); document.write( "

Algebra.Com's Answer #211967 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"t%5E2%2B3t-10\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-10\"$ to get $\"%281%29%28-10%29=-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-10\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\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 $\"-10\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-10\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,10\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10\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 $\"-10\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-10) = -10
\n" ); document.write( "2*(-5) = -10
\n" ); document.write( "(-1)*(10) = -10
\n" ); document.write( "(-2)*(5) = -10\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 $\"3\"$ :\r
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First Number	Second Number	Sum
1	-10	1+(-10)=-9
2	-5	2+(-5)=-3
-1	10	-1+10=9
-2	5	-2+5=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"5\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"5\"$ both multiply to $\"-10\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3t\"$ with $\"-2t%2B5t\"$ . Remember, $\"-2\"$ and $\"5\"$ add to $\"3\"$ . So this shows us that $\"-2t%2B5t=3t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"t%5E2%2Bhighlight%28-2t%2B5t%29-10\"$ Replace the second term $\"3t\"$ with $\"-2t%2B5t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28t%5E2-2t%29%2B%285t-10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t-2%29%2B%285t-10%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t-2%29%2B5%28t-2%29\"$ Factor out $\"5\"$ 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( " $\"%28t%2B5%29%28t-2%29\"$ Combine like terms. Or factor out the common term $\"t-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"t%5E2%2B3t-10\"$ factors to $\"%28t%2B5%29%28t-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"t%5E2%2B3t-10=%28t%2B5%29%28t-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28t%2B5%29%28t-2%29\"$ to get $\"t%5E2%2B3t-10\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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