document.write( "Question 153200: factor completly:\r
\n" ); document.write( "\n" ); document.write( "10x^3-31x+15 \n" ); document.write( "

Algebra.Com's Answer #112721 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Are you sure that the expression is not $\"10x%5E2-31x%2B15\"$ ???\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"10x%5E2-31x%2B15\"$ , we can see that the first coefficient is $\"10\"$ , the second coefficient is $\"-31\"$ , and the last term is $\"15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"10\"$ by the last term $\"15\"$ to get $\"%2810%29%2815%29=150\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"150\"$ (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 $\"150\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"150\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,25,30,50,75,150\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-25,-30,-50,-75,-150\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 $\"150\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*150
\n" ); document.write( "2*75
\n" ); document.write( "3*50
\n" ); document.write( "5*30
\n" ); document.write( "6*25
\n" ); document.write( "10*15
\n" ); document.write( "(-1)*(-150)
\n" ); document.write( "(-2)*(-75)
\n" ); document.write( "(-3)*(-50)
\n" ); document.write( "(-5)*(-30)
\n" ); document.write( "(-6)*(-25)
\n" ); document.write( "(-10)*(-15)\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	150	1+150=151
2	75	2+75=77
3	50	3+50=53
5	30	5+30=35
6	25	6+25=31
10	15	10+15=25
-1	-150	-1+(-150)=-151
-2	-75	-2+(-75)=-77
-3	-50	-3+(-50)=-53
-5	-30	-5+(-30)=-35
-6	-25	-6+(-25)=-31
-10	-15	-10+(-15)=-25

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"-25\"$ add to $\"-31\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"-25\"$ both multiply to $\"150\"$ and add to $\"-31\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-31x\"$ with $\"-6x-25x\"$ . Remember, $\"-6\"$ and $\"-25\"$ add to $\"-31\"$ . So this shows us that $\"-6x-25x=-31x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"10x%5E2%2Bhighlight%28-6x-25x%29%2B15\"$ Replace the second term $\"-31x\"$ with $\"-6x-25x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2810x%5E2-6x%29%2B%28-25x%2B15%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%285x-3%29%2B%28-25x%2B15%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%285x-3%29-5%285x-3%29\"$ 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( " $\"%282x-5%29%285x-3%29\"$ Combine like terms. Or factor out the common term $\"5x-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"10x%5E2-31x%2B15\"$ factors to $\"%282x-5%29%285x-3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x-5%29%285x-3%29\"$ to get $\"10x%5E2-31x%2B15\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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