document.write( "Question 641543: I need to completely factor 6x^3-31x^2+5x. \n" ); document.write( "

Algebra.Com's Answer #403718 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6x%5E3-31x%5E2%2B5x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%286x%5E2-31x%2B5%29\"$ 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 $\"6x%5E2-31x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2-31x%2B5\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-31\"$ , and the last term is $\"5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"5\"$ to get $\"%286%29%285%29=30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"30\"$ (the previous product) and add to the second coefficient $\"-31\"$ ?\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 $\"30\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"30\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\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 $\"30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*30 = 30
\n" ); document.write( "2*15 = 30
\n" ); document.write( "3*10 = 30
\n" ); document.write( "5*6 = 30
\n" ); document.write( "(-1)*(-30) = 30
\n" ); document.write( "(-2)*(-15) = 30
\n" ); document.write( "(-3)*(-10) = 30
\n" ); document.write( "(-5)*(-6) = 30\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 $\"-31\"$ :\r
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First Number	Second Number	Sum
1	30	1+30=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"-30\"$ add to $\"-31\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"-30\"$ both multiply to $\"30\"$ and add to $\"-31\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-31x\"$ with $\"-x-30x\"$ . Remember, $\"-1\"$ and $\"-30\"$ add to $\"-31\"$ . So this shows us that $\"-x-30x=-31x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%28-x-30x%29%2B5\"$ Replace the second term $\"-31x\"$ with $\"-x-30x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2-x%29%2B%28-30x%2B5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%286x-1%29%2B%28-30x%2B5%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%286x-1%29-5%286x-1%29\"$ Factor out $\"5\"$ 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( " $\"%28x-5%29%286x-1%29\"$ Combine like terms. Or factor out the common term $\"6x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%286x%5E2-31x%2B5%29\"$ then factors further to $\"x%28x-5%29%286x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E3-31x%5E2%2B5x\"$ completely factors to $\"x%28x-5%29%286x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E3-31x%5E2%2B5x=x%28x-5%29%286x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"x%28x-5%29%286x-1%29\"$ to get $\"6x%5E3-31x%5E2%2B5x\"$ 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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