document.write( "Question 207732: Dear sir or mam:
\n" ); document.write( "I am working on factoring trinomials completely
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Algebra.Com's Answer #157118 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"4p%5E2-16p-20\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"4%28p%5E2-4p-5%29\"$ Factor out the GCF $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"p%5E2-4p-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"p%5E2-4p-5\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-4\"$ , and the last term is $\"-5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-5\"$ to get $\"%281%29%28-5%29=-5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-5\"$ (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 $\"-5\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-5\"$ :\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 $\"-5\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-5) = -5
\n" ); document.write( "(-1)*(5) = -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 $\"-4\"$ :\r
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First Number	Second Number	Sum
1	-5	1+(-5)=-4
-1	5	-1+5=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"-5\"$ add to $\"-4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"-5\"$ both multiply to $\"-5\"$ and add to $\"-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4p\"$ with $\"p-5p\"$ . Remember, $\"1\"$ and $\"-5\"$ add to $\"-4\"$ . So this shows us that $\"p-5p=-4p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"p%5E2%2Bhighlight%28p-5p%29-5\"$ Replace the second term $\"-4p\"$ with $\"p-5p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28p%5E2%2Bp%29%2B%28-5p-5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%28p%2B1%29%2B%28-5p-5%29\"$ Factor out the GCF $\"p\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%28p%2B1%29-5%28p%2B1%29\"$ 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( " $\"%28p-5%29%28p%2B1%29\"$ Combine like terms. Or factor out the common term $\"p%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4%28p%5E2-4p-5%29\"$ then factors further to $\"4%28p-5%29%28p%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4p%5E2-16p-20\"$ completely factors to $\"4%28p-5%29%28p%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"4p%5E2-16p-20=4%28p-5%29%28p%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"4%28p-5%29%28p%2B1%29\"$ to get $\"4p%5E2-16p-20\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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