document.write( "Question 704242: factor completely\r
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Algebra.Com's Answer #433979 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"-15x%5E2-81x-30\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-3%285x%5E2%2B27x%2B10%29\"$ 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 $\"5x%5E2%2B27x%2B10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5x%5E2%2B27x%2B10\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"27\"$ , and the last term is $\"10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"10\"$ to get $\"%285%29%2810%29=50\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"50\"$ (the previous product) and add to the second coefficient $\"27\"$ ?\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 $\"50\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"50\"$ :\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 $\"50\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*50 = 50
\n" ); document.write( "2*25 = 50
\n" ); document.write( "5*10 = 50
\n" ); document.write( "(-1)*(-50) = 50
\n" ); document.write( "(-2)*(-25) = 50
\n" ); document.write( "(-5)*(-10) = 50\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 $\"27\"$ :\r
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First Number	Second Number	Sum
1	50	1+50=51
2	25	2+25=27
5	10	5+10=15
-1	-50	-1+(-50)=-51
-2	-25	-2+(-25)=-27
-5	-10	-5+(-10)=-15

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"25\"$ add to $\"27\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"25\"$ both multiply to $\"50\"$ and add to $\"27\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"27x\"$ with $\"2x%2B25x\"$ . Remember, $\"2\"$ and $\"25\"$ add to $\"27\"$ . So this shows us that $\"2x%2B25x=27x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2Bhighlight%282x%2B25x%29%2B10\"$ Replace the second term $\"27x\"$ with $\"2x%2B25x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%5E2%2B2x%29%2B%2825x%2B10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%285x%2B2%29%2B%2825x%2B10%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%285x%2B2%29%2B5%285x%2B2%29\"$ 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( " $\"%28x%2B5%29%285x%2B2%29\"$ Combine like terms. Or factor out the common term $\"5x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-3%285x%5E2%2B27x%2B10%29\"$ then factors further to $\"-3%28x%2B5%29%285x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-15x%5E2-81x-30\"$ completely factors to $\"-3%28x%2B5%29%285x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-15x%5E2-81x-30=-3%28x%2B5%29%285x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-3%28x%2B5%29%285x%2B2%29\"$ to get $\"-15x%5E2-81x-30\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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