document.write( "Question 610032: Factor completely: \r
\n" ); document.write( "\n" ); document.write( " 40x^3 + 34x^2 - 20x \n" ); document.write( "

Algebra.Com's Answer #384079 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"40x%5E3%2B34x%5E2-20x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%2820x%5E2%2B17x-10%29\"$ Factor out the GCF $\"2x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"20x%5E2%2B17x-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"20x%5E2%2B17x-10\"$ , we can see that the first coefficient is $\"20\"$ , the second coefficient is $\"17\"$ , and the last term is $\"-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"20\"$ by the last term $\"-10\"$ to get $\"%2820%29%28-10%29=-200\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-200\"$ (the previous product) and add to the second coefficient $\"17\"$ ?\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 $\"-200\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-200\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,25,40,50,100,200\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-25,-40,-50,-100,-200\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 $\"-200\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-200) = -200
\n" ); document.write( "2*(-100) = -200
\n" ); document.write( "4*(-50) = -200
\n" ); document.write( "5*(-40) = -200
\n" ); document.write( "8*(-25) = -200
\n" ); document.write( "10*(-20) = -200
\n" ); document.write( "(-1)*(200) = -200
\n" ); document.write( "(-2)*(100) = -200
\n" ); document.write( "(-4)*(50) = -200
\n" ); document.write( "(-5)*(40) = -200
\n" ); document.write( "(-8)*(25) = -200
\n" ); document.write( "(-10)*(20) = -200\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 $\"17\"$ :\r
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First Number	Second Number	Sum
1	-200	1+(-200)=-199
2	-100	2+(-100)=-98
4	-50	4+(-50)=-46
5	-40	5+(-40)=-35
8	-25	8+(-25)=-17
10	-20	10+(-20)=-10
-1	200	-1+200=199
-2	100	-2+100=98
-4	50	-4+50=46
-5	40	-5+40=35
-8	25	-8+25=17
-10	20	-10+20=10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-8\"$ and $\"25\"$ add to $\"17\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-8\"$ and $\"25\"$ both multiply to $\"-200\"$ and add to $\"17\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"17x\"$ with $\"-8x%2B25x\"$ . Remember, $\"-8\"$ and $\"25\"$ add to $\"17\"$ . So this shows us that $\"-8x%2B25x=17x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"20x%5E2%2Bhighlight%28-8x%2B25x%29-10\"$ Replace the second term $\"17x\"$ with $\"-8x%2B25x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2820x%5E2-8x%29%2B%2825x-10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%285x-2%29%2B%2825x-10%29\"$ Factor out the GCF $\"4x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%285x-2%29%2B5%285x-2%29\"$ 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( " $\"%284x%2B5%29%285x-2%29\"$ Combine like terms. Or factor out the common term $\"5x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%2820x%5E2%2B17x-10%29\"$ then factors further to $\"2x%284x%2B5%29%285x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"40x%5E3%2B34x%5E2-20x\"$ completely factors to $\"2x%284x%2B5%29%285x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"40x%5E3%2B34x%5E2-20x=2x%284x%2B5%29%285x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2x%284x%2B5%29%285x-2%29\"$ to get $\"40x%5E3%2B34x%5E2-20x\"$ 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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