document.write( "Question 338529: factor completely\r
\n" ); document.write( "\n" ); document.write( "(-x^3 +x^2 +42x) \n" ); document.write( "

Algebra.Com's Answer #242974 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"-x%5E3%2Bx%5E2%2B42x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-x%28x%5E2-x-42%29\"$ Factor out the GCF $\"-x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"x%5E2-x-42\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-x-42\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-42\"$ to get $\"%281%29%28-42%29=-42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-42\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?\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 $\"-42\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-42\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,14,21,42\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42\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 $\"-42\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-42) = -42
\n" ); document.write( "2*(-21) = -42
\n" ); document.write( "3*(-14) = -42
\n" ); document.write( "6*(-7) = -42
\n" ); document.write( "(-1)*(42) = -42
\n" ); document.write( "(-2)*(21) = -42
\n" ); document.write( "(-3)*(14) = -42
\n" ); document.write( "(-6)*(7) = -42\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 $\"-1\"$ :\r
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First Number	Second Number	Sum
1	-42	1+(-42)=-41
2	-21	2+(-21)=-19
3	-14	3+(-14)=-11
6	-7	6+(-7)=-1
-1	42	-1+42=41
-2	21	-2+21=19
-3	14	-3+14=11
-6	7	-6+7=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"6\"$ and $\"-7\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"-7\"$ both multiply to $\"-42\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1x\"$ with $\"6x-7x\"$ . Remember, $\"6\"$ and $\"-7\"$ add to $\"-1\"$ . So this shows us that $\"6x-7x=-1x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%286x-7x%29-42\"$ Replace the second term $\"-1x\"$ with $\"6x-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B6x%29%2B%28-7x-42%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B6%29%2B%28-7x-42%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B6%29-7%28x%2B6%29\"$ Factor out $\"7\"$ 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( " $\"%28x-7%29%28x%2B6%29\"$ Combine like terms. Or factor out the common term $\"x%2B6\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-1x%28x%5E2-x-42%29\"$ then factors further to $\"-%28x-7%29%28x%2B6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-x%5E3%2Bx%5E2%2B42x\"$ completely factors to $\"-%28x-7%29%28x%2B6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-x%5E3%2Bx%5E2%2B42x=-x%28x-7%29%28x%2B6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-x%28x-7%29%28x%2B6%29\"$ to get $\"-x%5E3%2Bx%5E2%2B42x\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website. \r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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