document.write( "Question 928275: Please show the steps to factor completely
\n" ); document.write( "6x^3 +5x^2 -4x \n" ); document.write( "

Algebra.Com's Answer #563554 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6x%5E3%2B5x%5E2-4x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%286x%5E2%2B5x-4%29\"$ 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 $\"6x%5E2%2B5x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2%2B5x-4\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-4\"$ to get $\"%286%29%28-4%29=-24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-24\"$ (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 $\"-24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"-24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-24) = -24
\n" ); document.write( "2*(-12) = -24
\n" ); document.write( "3*(-8) = -24
\n" ); document.write( "4*(-6) = -24
\n" ); document.write( "(-1)*(24) = -24
\n" ); document.write( "(-2)*(12) = -24
\n" ); document.write( "(-3)*(8) = -24
\n" ); document.write( "(-4)*(6) = -24\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	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"8\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"8\"$ both multiply to $\"-24\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5x\"$ with $\"-3x%2B8x\"$ . Remember, $\"-3\"$ and $\"8\"$ add to $\"5\"$ . So this shows us that $\"-3x%2B8x=5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%28-3x%2B8x%29-4\"$ Replace the second term $\"5x\"$ with $\"-3x%2B8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2-3x%29%2B%288x-4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x-1%29%2B%288x-4%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x-1%29%2B4%282x-1%29\"$ 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( " $\"%283x%2B4%29%282x-1%29\"$ Combine like terms. Or factor out the common term $\"2x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%286x%5E2%2B5x-4%29\"$ then factors further to $\"x%283x%2B4%29%282x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E3%2B5x%5E2-4x\"$ completely factors to $\"x%283x%2B4%29%282x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E3%2B5x%5E2-4x=x%283x%2B4%29%282x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"x%283x%2B4%29%282x-1%29\"$ to get $\"6x%5E3%2B5x%5E2-4x\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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\n" ); document.write( "\n" ); document.write( "If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).\r
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\n" ); document.write( "\n" ); document.write( "Alternatively, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.\r
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\n" ); document.write( "\n" ); document.write( "Jim\r
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