document.write( "Question 319179: Factor the following polynomial completely. (If the polynomial cannot be factored, enter PRIME.) \r
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Algebra.Com's Answer #228527 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"-x%5E2%2B16x%2B57\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-%28x%5E2-16x-57%29\"$ Factor out the GCF $\"-1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"x%5E2-16x-57\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-16x-57\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-16\"$ , and the last term is $\"-57\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-57\"$ to get $\"%281%29%28-57%29=-57\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-57\"$ (the previous product) and add to the second coefficient $\"-16\"$ ?\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 $\"-57\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-57\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-19,-57\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 $\"-57\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-57) = -57
\n" ); document.write( "3*(-19) = -57
\n" ); document.write( "(-1)*(57) = -57
\n" ); document.write( "(-3)*(19) = -57\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 $\"-16\"$ :\r
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First Number	Second Number	Sum
1	-57	1+(-57)=-56
3	-19	3+(-19)=-16
-1	57	-1+57=56
-3	19	-3+19=16

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-19\"$ add to $\"-16\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-19\"$ both multiply to $\"-57\"$ and add to $\"-16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-16x\"$ with $\"3x-19x\"$ . Remember, $\"3\"$ and $\"-19\"$ add to $\"-16\"$ . So this shows us that $\"3x-19x=-16x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283x-19x%29-57\"$ Replace the second term $\"-16x\"$ with $\"3x-19x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3x%29%2B%28-19x-57%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B%28-19x-57%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29-19%28x%2B3%29\"$ Factor out $\"19\"$ 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( " $\"%28x-19%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-1%28x%5E2-16x-57%29\"$ then factors further to $\"-%28x-19%29%28x%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-x%5E2%2B16x%2B57\"$ completely factors to $\"-%28x-19%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-x%5E2%2B16x%2B57=-%28x-19%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-%28x-19%29%28x%2B3%29\"$ to get $\"-x%5E2%2B16x%2B57\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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