document.write( "Question 908042: Factor the expressions:
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Algebra.Com's Answer #550762 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"-r%5E2%2B11r-18\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-%28r%5E2-11r%2B18%29\"$ 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 $\"r%5E2-11r%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"r%5E2-11r%2B18\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-11\"$ , and the last term is $\"18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"18\"$ to get $\"%281%29%2818%29=18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"18\"$ (the previous product) and add to the second coefficient $\"-11\"$ ?\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 $\"18\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"18\"$ :\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 $\"18\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*18 = 18
\n" ); document.write( "2*9 = 18
\n" ); document.write( "3*6 = 18
\n" ); document.write( "(-1)*(-18) = 18
\n" ); document.write( "(-2)*(-9) = 18
\n" ); document.write( "(-3)*(-6) = 18\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 $\"-11\"$ :\r
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First Number	Second Number	Sum
1	18	1+18=19
2	9	2+9=11
3	6	3+6=9
-1	-18	-1+(-18)=-19
-2	-9	-2+(-9)=-11
-3	-6	-3+(-6)=-9

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-9\"$ add to $\"-11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-9\"$ both multiply to $\"18\"$ and add to $\"-11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-11r\"$ with $\"-2r-9r\"$ . Remember, $\"-2\"$ and $\"-9\"$ add to $\"-11\"$ . So this shows us that $\"-2r-9r=-11r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%28-2r-9r%29%2B18\"$ Replace the second term $\"-11r\"$ with $\"-2r-9r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2-2r%29%2B%28-9r%2B18%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-2%29%2B%28-9r%2B18%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-2%29-9%28r-2%29\"$ Factor out $\"9\"$ 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( " $\"%28r-9%29%28r-2%29\"$ Combine like terms. Or factor out the common term $\"r-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-1%28r%5E2-11r%2B18%29\"$ then factors further to $\"-%28r-9%29%28r-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-r%5E2%2B11r-18\"$ completely factors to $\"-%28r-9%29%28r-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-r%5E2%2B11r-18=-%28r-9%29%28r-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-%28r-9%29%28r-2%29\"$ to get $\"-r%5E2%2B11r-18\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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