document.write( "Question 640850: Factor the following:
\n" ); document.write( "32x^2-80x+50\r
\n" ); document.write( "\n" ); document.write( "Xa+ya+x+y\r
\n" ); document.write( "\n" ); document.write( "6xy-8x+15y-20\r
\n" ); document.write( "\n" ); document.write( "30x^2y+35x^2y^2\r
\n" ); document.write( "\n" ); document.write( "12a^2-15ab-16a+20b\r
\n" ); document.write( "\n" ); document.write( "4x^2+26x-48\r
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Algebra.Com's Answer #403437 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first one to get you started.\r
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\n" ); document.write( "\n" ); document.write( " $\"32x%5E2-80x%2B50\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2816x%5E2-40x%2B25%29\"$ Factor out the GCF $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"16x%5E2-40x%2B25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"16x%5E2-40x%2B25\"$ , we can see that the first coefficient is $\"16\"$ , the second coefficient is $\"-40\"$ , and the last term is $\"25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"16\"$ by the last term $\"25\"$ to get $\"%2816%29%2825%29=400\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"400\"$ (the previous product) and add to the second coefficient $\"-40\"$ ?\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 $\"400\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"400\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,25,40,50,80,100,200,400\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200,-400\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 $\"400\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*400 = 400
\n" ); document.write( "2*200 = 400
\n" ); document.write( "4*100 = 400
\n" ); document.write( "5*80 = 400
\n" ); document.write( "8*50 = 400
\n" ); document.write( "10*40 = 400
\n" ); document.write( "16*25 = 400
\n" ); document.write( "20*20 = 400
\n" ); document.write( "(-1)*(-400) = 400
\n" ); document.write( "(-2)*(-200) = 400
\n" ); document.write( "(-4)*(-100) = 400
\n" ); document.write( "(-5)*(-80) = 400
\n" ); document.write( "(-8)*(-50) = 400
\n" ); document.write( "(-10)*(-40) = 400
\n" ); document.write( "(-16)*(-25) = 400
\n" ); document.write( "(-20)*(-20) = 400\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 $\"-40\"$ :\r
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First Number	Second Number	Sum
1	400	1+400=401
2	200	2+200=202
4	100	4+100=104
5	80	5+80=85
8	50	8+50=58
10	40	10+40=50
16	25	16+25=41
20	20	20+20=40
-1	-400	-1+(-400)=-401
-2	-200	-2+(-200)=-202
-4	-100	-4+(-100)=-104
-5	-80	-5+(-80)=-85
-8	-50	-8+(-50)=-58
-10	-40	-10+(-40)=-50
-16	-25	-16+(-25)=-41
-20	-20	-20+(-20)=-40

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-20\"$ and $\"-20\"$ add to $\"-40\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-20\"$ and $\"-20\"$ both multiply to $\"400\"$ and add to $\"-40\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-40x\"$ with $\"-20x-20x\"$ . Remember, $\"-20\"$ and $\"-20\"$ add to $\"-40\"$ . So this shows us that $\"-20x-20x=-40x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"16x%5E2%2Bhighlight%28-20x-20x%29%2B25\"$ Replace the second term $\"-40x\"$ with $\"-20x-20x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2816x%5E2-20x%29%2B%28-20x%2B25%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%284x-5%29%2B%28-20x%2B25%29\"$ Factor out the GCF $\"4x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%284x-5%29-5%284x-5%29\"$ 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( " $\"%284x-5%29%284x-5%29\"$ Combine like terms. Or factor out the common term $\"4x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%284x-5%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"2%2816x%5E2-40x%2B25%29\"$ then factors further to $\"2%284x-5%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"32x%5E2-80x%2B50\"$ completely factors to $\"2%284x-5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"32x%5E2-80x%2B50=2%284x-5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%284x-5%29%5E2\"$ to get $\"32x%5E2-80x%2B50\"$ 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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