document.write( "Question 190287: Factor\r
\n" ); document.write( "\n" ); document.write( "80+36t+4t^2 \n" ); document.write( "

Algebra.Com's Answer #142805 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"80%2B36t%2B4t%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"4t%5E2%2B36t%2B80\"$ Rearrange the terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"4%28t%5E2%2B9t%2B20%29\"$ Factor out the GCF $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"t%5E2%2B9t%2B20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"t%5E2%2B9t%2B20\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"9\"$ , and the last term is $\"20\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"20\"$ to get $\"%281%29%2820%29=20\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"20\"$ (the previous product) and add to the second coefficient $\"9\"$ ?\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 $\"20\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"20\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20\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 $\"20\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*20
\n" ); document.write( "2*10
\n" ); document.write( "4*5
\n" ); document.write( "(-1)*(-20)
\n" ); document.write( "(-2)*(-10)
\n" ); document.write( "(-4)*(-5)\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 $\"9\"$ :\r
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First Number	Second Number	Sum
1	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"4\"$ and $\"5\"$ add to $\"9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"5\"$ both multiply to $\"20\"$ and add to $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"9t\"$ with $\"4t%2B5t\"$ . Remember, $\"4\"$ and $\"5\"$ add to $\"9\"$ . So this shows us that $\"4t%2B5t=9t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"t%5E2%2Bhighlight%284t%2B5t%29%2B20\"$ Replace the second term $\"9t\"$ with $\"4t%2B5t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28t%5E2%2B4t%29%2B%285t%2B20%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t%2B4%29%2B%285t%2B20%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%28t%2B4%29%2B5%28t%2B4%29\"$ 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( " $\"%28t%2B5%29%28t%2B4%29\"$ Combine like terms. Or factor out the common term $\"t%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"t%5E2%2B9t%2B20\"$ factors to $\"%28t%2B5%29%28t%2B4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4%28t%5E2%2B9t%2B20%29\"$ and factors further to $\"4%28t%2B5%29%28t%2B4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"80%2B36t%2B4t%5E2\"$ completely factors to $\"4%28t%2B5%29%28t%2B4%29\"$ \n" ); document.write( "

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