document.write( "Question 538206: factor x^4-11x^3-12x^2
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Algebra.Com's Answer #353180 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"x%5E4-11x%5E3-12x%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%28x%5E2-11x-12%29\"$ Factor out the GCF $\"x%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"x%5E2-11x-12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-11x-12\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-11\"$ , and the last term is $\"-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-12\"$ to get $\"%281%29%28-12%29=-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-12\"$ (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 $\"-12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-12\"$ :\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 $\"-12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-12) = -12
\n" ); document.write( "2*(-6) = -12
\n" ); document.write( "3*(-4) = -12
\n" ); document.write( "(-1)*(12) = -12
\n" ); document.write( "(-2)*(6) = -12
\n" ); document.write( "(-3)*(4) = -12\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	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"-12\"$ add to $\"-11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"-12\"$ both multiply to $\"-12\"$ and add to $\"-11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-11x\"$ with $\"x-12x\"$ . Remember, $\"1\"$ and $\"-12\"$ add to $\"-11\"$ . So this shows us that $\"x-12x=-11x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28x-12x%29-12\"$ Replace the second term $\"-11x\"$ with $\"x-12x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2Bx%29%2B%28-12x-12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B1%29%2B%28-12x-12%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B1%29-12%28x%2B1%29\"$ Factor out $\"12\"$ 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( " $\"%28x-12%29%28x%2B1%29\"$ Combine like terms. Or factor out the common term $\"x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%28x%5E2-11x-12%29\"$ then factors further to $\"x%5E2%28x-12%29%28x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E4-11x%5E3-12x%5E2\"$ completely factors to $\"x%5E2%28x-12%29%28x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E4-11x%5E3-12x%5E2=x%5E2%28x-12%29%28x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"x%5E2%28x-12%29%28x%2B1%29\"$ to get $\"x%5E4-11x%5E3-12x%5E2\"$ 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, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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