document.write( "Question 190250: Factor the trinomial\r
\n" ); document.write( "\n" ); document.write( "3x^2+27x+42 \n" ); document.write( "

Algebra.Com's Answer #142770 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2B27x%2B42\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28x%5E2%2B9x%2B14%29\"$ Factor out the GCF $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2%2B9x%2B14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B9x%2B14\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"9\"$ , and the last term is $\"14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"14\"$ to get $\"%281%29%2814%29=14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"14\"$ (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 $\"14\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"14\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14\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 $\"14\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*14
\n" ); document.write( "2*7
\n" ); document.write( "(-1)*(-14)
\n" ); document.write( "(-2)*(-7)\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	14	1+14=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"7\"$ add to $\"9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"7\"$ both multiply to $\"14\"$ and add to $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"9x\"$ with $\"2x%2B7x\"$ . Remember, $\"2\"$ and $\"7\"$ add to $\"9\"$ . So this shows us that $\"2x%2B7x=9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%282x%2B7x%29%2B14\"$ Replace the second term $\"9x\"$ with $\"2x%2B7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2x%29%2B%287x%2B14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B%287x%2B14%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B7%28x%2B2%29\"$ Factor out $\"7\"$ 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( " $\"%28x%2B7%29%28x%2B2%29\"$ Combine like terms. Or factor out the common term $\"x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B9x%2B14\"$ factors to $\"%28x%2B7%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"3%28x%5E2%2B9x%2B14%29\"$ and factors further to $\"3%28x%2B7%29%28x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B27x%2B42\"$ completely factors to $\"3%28x%2B7%29%28x%2B2%29\"$ \r
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