document.write( "Question 318015: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. \r
\n" ); document.write( "\n" ); document.write( "3c^2 + 6c - 105 \n" ); document.write( "

Algebra.Com's Answer #227703 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"3c%5E2%2B6c-105\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28c%5E2%2B2c-35%29\"$ 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 $\"c%5E2%2B2c-35\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"c%5E2%2B2c-35\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-35\"$ to get $\"%281%29%28-35%29=-35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-35\"$ (the previous product) and add to the second coefficient $\"2\"$ ?\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 $\"-35\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-35\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,5,7,35\r
\n" ); document.write( "\n" ); document.write( "-1,-5,-7,-35\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 $\"-35\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-35) = -35
\n" ); document.write( "5*(-7) = -35
\n" ); document.write( "(-1)*(35) = -35
\n" ); document.write( "(-5)*(7) = -35\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 $\"2\"$ :\r
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First Number	Second Number	Sum
1	-35	1+(-35)=-34
5	-7	5+(-7)=-2
-1	35	-1+35=34
-5	7	-5+7=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"7\"$ add to $\"2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"7\"$ both multiply to $\"-35\"$ and add to $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"2c\"$ with $\"-5c%2B7c\"$ . Remember, $\"-5\"$ and $\"7\"$ add to $\"2\"$ . So this shows us that $\"-5c%2B7c=2c\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"c%5E2%2Bhighlight%28-5c%2B7c%29-35\"$ Replace the second term $\"2c\"$ with $\"-5c%2B7c\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28c%5E2-5c%29%2B%287c-35%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"c%28c-5%29%2B%287c-35%29\"$ Factor out the GCF $\"c\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"c%28c-5%29%2B7%28c-5%29\"$ 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( " $\"%28c%2B7%29%28c-5%29\"$ Combine like terms. Or factor out the common term $\"c-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3%28c%5E2%2B2c-35%29\"$ then factors further to $\"3%28c%2B7%29%28c-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3c%5E2%2B6c-105\"$ completely factors to $\"3%28c%2B7%29%28c-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3c%5E2%2B6c-105=3%28c%2B7%29%28c-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%28c%2B7%29%28c-5%29\"$ to get $\"3c%5E2%2B6c-105\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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