document.write( "Question 530360: factor the following trinomial completely: 3x^2+12x+12
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Algebra.Com's Answer #350017 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2B12x%2B12\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28x%5E2%2B4x%2B4%29\"$ 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 $\"x%5E2%2B4x%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B4x%2B4\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"4\"$ , and the last term is $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"4\"$ to get $\"%281%29%284%29=4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"4\"$ (the previous product) and add to the second coefficient $\"4\"$ ?\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 $\"4\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"4\"$ :\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 $\"4\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*4 = 4
\n" ); document.write( "2*2 = 4
\n" ); document.write( "(-1)*(-4) = 4
\n" ); document.write( "(-2)*(-2) = 4\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 $\"4\"$ :\r
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First Number	Second Number	Sum
1	4	1+4=5
2	2	2+2=4
-1	-4	-1+(-4)=-5
-2	-2	-2+(-2)=-4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"2\"$ add to $\"4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"2\"$ both multiply to $\"4\"$ and add to $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"4x\"$ with $\"2x%2B2x\"$ . Remember, $\"2\"$ and $\"2\"$ add to $\"4\"$ . So this shows us that $\"2x%2B2x=4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%282x%2B2x%29%2B4\"$ Replace the second term $\"4x\"$ with $\"2x%2B2x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2x%29%2B%282x%2B4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B%282x%2B4%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B2%28x%2B2%29\"$ Factor out $\"2\"$ 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( " $\"%28x%2B2%29%28x%2B2%29\"$ Combine like terms. Or factor out the common term $\"x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B2%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"3%28x%5E2%2B4x%2B4%29\"$ then factors further to $\"3%28x%2B2%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B12x%2B12\"$ completely factors to $\"3%28x%2B2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2%2B12x%2B12=3%28x%2B2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%28x%2B2%29%5E2\"$ to get $\"3x%5E2%2B12x%2B12\"$ 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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