document.write( "Question 521778: factor trinomial using trial and error\r
\n" ); document.write( "\n" ); document.write( "6x^2+39x+60\r
\n" ); document.write( "\n" ); document.write( "or is it not factorable?
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Algebra.Com's Answer #346575 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2B39x%2B60\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%282x%5E2%2B13x%2B20%29\"$ Factor out the GCF $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"2x%5E2%2B13x%2B20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B13x%2B20\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"13\"$ , and the last term is $\"20\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"20\"$ to get $\"%282%29%2820%29=40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"40\"$ (the previous product) and add to the second coefficient $\"13\"$ ?\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 $\"40\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"40\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40\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 $\"40\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*40 = 40
\n" ); document.write( "2*20 = 40
\n" ); document.write( "4*10 = 40
\n" ); document.write( "5*8 = 40
\n" ); document.write( "(-1)*(-40) = 40
\n" ); document.write( "(-2)*(-20) = 40
\n" ); document.write( "(-4)*(-10) = 40
\n" ); document.write( "(-5)*(-8) = 40\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 $\"13\"$ :\r
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First Number	Second Number	Sum
1	40	1+40=41
2	20	2+20=22
4	10	4+10=14
5	8	5+8=13
-1	-40	-1+(-40)=-41
-2	-20	-2+(-20)=-22
-4	-10	-4+(-10)=-14
-5	-8	-5+(-8)=-13

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"8\"$ add to $\"13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"8\"$ both multiply to $\"40\"$ and add to $\"13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"13x\"$ with $\"5x%2B8x\"$ . Remember, $\"5\"$ and $\"8\"$ add to $\"13\"$ . So this shows us that $\"5x%2B8x=13x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%285x%2B8x%29%2B20\"$ Replace the second term $\"13x\"$ with $\"5x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2%2B5x%29%2B%288x%2B20%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B5%29%2B%288x%2B20%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B5%29%2B4%282x%2B5%29\"$ Factor out $\"4\"$ 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( " $\"%28x%2B4%29%282x%2B5%29\"$ Combine like terms. Or factor out the common term $\"2x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3%282x%5E2%2B13x%2B20%29\"$ then factors further to $\"3%28x%2B4%29%282x%2B5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2%2B39x%2B60\"$ completely factors to $\"3%28x%2B4%29%282x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E2%2B39x%2B60=3%28x%2B4%29%282x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%28x%2B4%29%282x%2B5%29\"$ to get $\"6x%5E2%2B39x%2B60\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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\n" ); document.write( "\n" ); document.write( "Jim
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