document.write( "Question 166139: I could use some help with factoring this polynomial,
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Algebra.Com's Answer #122423 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%281%2F9%29x%5E2%2B%282%2F3%29x-63\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%281%2F9%29%28x%5E2%2B6x-567%29\"$ Factor out the GCF $\"1%2F9\"$ . This will make every term on the inside of the parenthesis have a whole coefficient.\r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2%2B6x-567\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B6x-567\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"6\"$ , and the last term is $\"-567\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-567\"$ to get $\"%281%29%28-567%29=-567\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-567\"$ (the previous product) and add to the second coefficient $\"6\"$ ?\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 $\"-567\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-567\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-27,-63,-81,-189,-567\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 $\"-567\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-567)
\n" ); document.write( "3*(-189)
\n" ); document.write( "7*(-81)
\n" ); document.write( "9*(-63)
\n" ); document.write( "21*(-27)
\n" ); document.write( "(-1)*(567)
\n" ); document.write( "(-3)*(189)
\n" ); document.write( "(-7)*(81)
\n" ); document.write( "(-9)*(63)
\n" ); document.write( "(-21)*(27)\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 $\"6\"$ :\r
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First Number	Second Number	Sum
1	-567	1+(-567)=-566
3	-189	3+(-189)=-186
7	-81	7+(-81)=-74
9	-63	9+(-63)=-54
21	-27	21+(-27)=-6
-1	567	-1+567=566
-3	189	-3+189=186
-7	81	-7+81=74
-9	63	-9+63=54
-21	27	-21+27=6

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-21\"$ and $\"27\"$ add to $\"6\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-21\"$ and $\"27\"$ both multiply to $\"-567\"$ and add to $\"6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"6x\"$ with $\"-21x%2B27x\"$ . Remember, $\"-21\"$ and $\"27\"$ add to $\"6\"$ . So this shows us that $\"-21x%2B27x=6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-21x%2B27x%29-567\"$ Replace the second term $\"6x\"$ with $\"-21x%2B27x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-21x%29%2B%2827x-567%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-21%29%2B%2827x-567%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-21%29%2B27%28x-21%29\"$ Factor out $\"27\"$ 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( " $\"%28x%2B27%29%28x-21%29\"$ Combine like terms. Or factor out the common term $\"x-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B6x-567\"$ factors to $\"%28x%2B27%29%28x-21%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"%281%2F9%29%28x%5E2%2B6x-567%29\"$ becomes $\"%281%2F9%29%28x%2B27%29%28x-21%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"%281%2F9%29x%5E2%2B%282%2F3%29x-63\"$ completely factors to $\"%281%2F9%29%28x%2B27%29%28x-21%29\"$ \n" ); document.write( "

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