document.write( "Question 913154: how do you factor \"x%5E2+-22x%2B120\" \n" ); document.write( "
Algebra.Com's Answer #554328 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "Looking at the expression \"x%5E2-22x%2B120\", we can see that the first coefficient is \"1\", the second coefficient is \"-22\", and the last term is \"120\".\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient \"1\" by the last term \"120\" to get \"%281%29%28120%29=120\".\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to \"120\" (the previous product) and add to the second coefficient \"-22\"?\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 \"120\" (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of \"120\":\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120\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 \"120\".\r
\n" ); document.write( "\n" ); document.write( "1*120 = 120
\n" ); document.write( "2*60 = 120
\n" ); document.write( "3*40 = 120
\n" ); document.write( "4*30 = 120
\n" ); document.write( "5*24 = 120
\n" ); document.write( "6*20 = 120
\n" ); document.write( "8*15 = 120
\n" ); document.write( "10*12 = 120
\n" ); document.write( "(-1)*(-120) = 120
\n" ); document.write( "(-2)*(-60) = 120
\n" ); document.write( "(-3)*(-40) = 120
\n" ); document.write( "(-4)*(-30) = 120
\n" ); document.write( "(-5)*(-24) = 120
\n" ); document.write( "(-6)*(-20) = 120
\n" ); document.write( "(-8)*(-15) = 120
\n" ); document.write( "(-10)*(-12) = 120\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 \"-22\":\r
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First NumberSecond NumberSum
11201+120=121
2602+60=62
3403+40=43
4304+30=34
5245+24=29
6206+20=26
8158+15=23
101210+12=22
-1-120-1+(-120)=-121
-2-60-2+(-60)=-62
-3-40-3+(-40)=-43
-4-30-4+(-30)=-34
-5-24-5+(-24)=-29
-6-20-6+(-20)=-26
-8-15-8+(-15)=-23
-10-12-10+(-12)=-22
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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers \"-10\" and \"-12\" add to \"-22\" (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers \"-10\" and \"-12\" both multiply to \"120\" and add to \"-22\"\r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term \"-22x\" with \"-10x-12x\". Remember, \"-10\" and \"-12\" add to \"-22\". So this shows us that \"-10x-12x=-22x\".\r
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\n" ); document.write( "\n" ); document.write( "\"x%5E2%2Bhighlight%28-10x-12x%29%2B120\" Replace the second term \"-22x\" with \"-10x-12x\".\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%5E2-10x%29%2B%28-12x%2B120%29\" Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( "\"x%28x-10%29%2B%28-12x%2B120%29\" Factor out the GCF \"x\" from the first group.\r
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\n" ); document.write( "\n" ); document.write( "\"x%28x-10%29-12%28x-10%29\" Factor out \"12\" 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-12%29%28x-10%29\" Combine like terms. Or factor out the common term \"x-10\"\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So \"x%5E2-22x%2B120\" factors to \"%28x-12%29%28x-10%29\".\r
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\n" ); document.write( "\n" ); document.write( "In other words, \"x%5E2-22x%2B120=%28x-12%29%28x-10%29\".\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding \"%28x-12%29%28x-10%29\" to get \"x%5E2-22x%2B120\" or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "Let me know if you need more help or if you need me to explain a step in more detail.
\n" ); document.write( "Feel free to email me at jim_thompson5910@hotmail.com
\n" ); document.write( "or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html\r
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\n" ); document.write( "\n" ); document.write( "Thanks,\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "
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