document.write( "Question 318721: 3x squared minus 6x minus 24.............im being asked to factor completely \n" ); document.write( "

Algebra.Com's Answer #228151 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2-6x-24\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28x%5E2-2x-8%29\"$ 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 $\"x%5E2-2x-8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-2x-8\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-8\"$ to get $\"%281%29%28-8%29=-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-8\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\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 $\"-8\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-8\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8\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 $\"-8\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-8) = -8
\n" ); document.write( "2*(-4) = -8
\n" ); document.write( "(-1)*(8) = -8
\n" ); document.write( "(-2)*(4) = -8\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 $\"-2\"$ :\r
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First Number	Second Number	Sum
1	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-2+4=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-4\"$ add to $\"-2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-4\"$ both multiply to $\"-8\"$ and add to $\"-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2x\"$ with $\"2x-4x\"$ . Remember, $\"2\"$ and $\"-4\"$ add to $\"-2\"$ . So this shows us that $\"2x-4x=-2x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%282x-4x%29-8\"$ Replace the second term $\"-2x\"$ with $\"2x-4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2x%29%2B%28-4x-8%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29%2B%28-4x-8%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29-4%28x%2B2%29\"$ Factor out $\"4\"$ 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( " $\"%28x-4%29%28x%2B2%29\"$ Combine like terms. Or factor out the common term $\"x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3%28x%5E2-2x-8%29\"$ then factors further to $\"3%28x-4%29%28x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-6x-24\"$ completely factors to $\"3%28x-4%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2-6x-24=3%28x-4%29%28x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%28x-4%29%28x%2B2%29\"$ to get $\"3x%5E2-6x-24\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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