document.write( "Question 653914: i want to know how to factor 48x^3 - 40x^2 - 48x \n" ); document.write( "

Algebra.Com's Answer #408548 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"48x%5E3-40x%5E2-48x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"8x%286x%5E2-5x-6%29\"$ Factor out the GCF $\"8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"6x%5E2-5x-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2-5x-6\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-6\"$ to get $\"%286%29%28-6%29=-36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-36\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\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 $\"-36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-36\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,36\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\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 $\"-36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-36) = -36
\n" ); document.write( "2*(-18) = -36
\n" ); document.write( "3*(-12) = -36
\n" ); document.write( "4*(-9) = -36
\n" ); document.write( "6*(-6) = -36
\n" ); document.write( "(-1)*(36) = -36
\n" ); document.write( "(-2)*(18) = -36
\n" ); document.write( "(-3)*(12) = -36
\n" ); document.write( "(-4)*(9) = -36
\n" ); document.write( "(-6)*(6) = -36\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 $\"-5\"$ :\r
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First Number	Second Number	Sum
1	-36	1+(-36)=-35
2	-18	2+(-18)=-16
3	-12	3+(-12)=-9
4	-9	4+(-9)=-5
6	-6	6+(-6)=0
-1	36	-1+36=35
-2	18	-2+18=16
-3	12	-3+12=9
-4	9	-4+9=5
-6	6	-6+6=0

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"4\"$ and $\"-9\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"-9\"$ both multiply to $\"-36\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5x\"$ with $\"4x-9x\"$ . Remember, $\"4\"$ and $\"-9\"$ add to $\"-5\"$ . So this shows us that $\"4x-9x=-5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%284x-9x%29-6\"$ Replace the second term $\"-5x\"$ with $\"4x-9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2%2B4x%29%2B%28-9x-6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x%2B2%29%2B%28-9x-6%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x%2B2%29-3%283x%2B2%29\"$ Factor out $\"3\"$ 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( " $\"%282x-3%29%283x%2B2%29\"$ Combine like terms. Or factor out the common term $\"3x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"8x%286x%5E2-5x-6%29\"$ then factors further to $\"8x%282x-3%29%283x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"48x%5E3-40x%5E2-48x\"$ completely factors to $\"8x%282x-3%29%283x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"48x%5E3-40x%5E2-48x=8x%282x-3%29%283x%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"8x%282x-3%29%283x%2B2%29\"$ to get $\"48x%5E3-40x%5E2-48x\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\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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