document.write( "Question 481201: 135x^2-222x+91\r
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Algebra.Com's Answer #329600 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"135x%5E2-222x%2B91\"$ , we can see that the first coefficient is $\"135\"$ , the second coefficient is $\"-222\"$ , and the last term is $\"91\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"135\"$ by the last term $\"91\"$ to get $\"%28135%29%2891%29=12285\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"12285\"$ (the previous product) and add to the second coefficient $\"-222\"$ ?\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 $\"12285\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"12285\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,7,9,13,15,21,27,35,39,45,63,65,91,105,117,135,189,195,273,315,351,455,585,819,945,1365,1755,2457,4095,12285\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-7,-9,-13,-15,-21,-27,-35,-39,-45,-63,-65,-91,-105,-117,-135,-189,-195,-273,-315,-351,-455,-585,-819,-945,-1365,-1755,-2457,-4095,-12285\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 $\"12285\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*12285 = 12285
\n" ); document.write( "3*4095 = 12285
\n" ); document.write( "5*2457 = 12285
\n" ); document.write( "7*1755 = 12285
\n" ); document.write( "9*1365 = 12285
\n" ); document.write( "13*945 = 12285
\n" ); document.write( "15*819 = 12285
\n" ); document.write( "21*585 = 12285
\n" ); document.write( "27*455 = 12285
\n" ); document.write( "35*351 = 12285
\n" ); document.write( "39*315 = 12285
\n" ); document.write( "45*273 = 12285
\n" ); document.write( "63*195 = 12285
\n" ); document.write( "65*189 = 12285
\n" ); document.write( "91*135 = 12285
\n" ); document.write( "105*117 = 12285
\n" ); document.write( "(-1)*(-12285) = 12285
\n" ); document.write( "(-3)*(-4095) = 12285
\n" ); document.write( "(-5)*(-2457) = 12285
\n" ); document.write( "(-7)*(-1755) = 12285
\n" ); document.write( "(-9)*(-1365) = 12285
\n" ); document.write( "(-13)*(-945) = 12285
\n" ); document.write( "(-15)*(-819) = 12285
\n" ); document.write( "(-21)*(-585) = 12285
\n" ); document.write( "(-27)*(-455) = 12285
\n" ); document.write( "(-35)*(-351) = 12285
\n" ); document.write( "(-39)*(-315) = 12285
\n" ); document.write( "(-45)*(-273) = 12285
\n" ); document.write( "(-63)*(-195) = 12285
\n" ); document.write( "(-65)*(-189) = 12285
\n" ); document.write( "(-91)*(-135) = 12285
\n" ); document.write( "(-105)*(-117) = 12285\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 $\"-222\"$ :\r
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First Number	Second Number	Sum
1	12285	1+12285=12286
3	4095	3+4095=4098
5	2457	5+2457=2462
7	1755	7+1755=1762
9	1365	9+1365=1374
13	945	13+945=958
15	819	15+819=834
21	585	21+585=606
27	455	27+455=482
35	351	35+351=386
39	315	39+315=354
45	273	45+273=318
63	195	63+195=258
65	189	65+189=254
91	135	91+135=226
105	117	105+117=222
-1	-12285	-1+(-12285)=-12286
-3	-4095	-3+(-4095)=-4098
-5	-2457	-5+(-2457)=-2462
-7	-1755	-7+(-1755)=-1762
-9	-1365	-9+(-1365)=-1374
-13	-945	-13+(-945)=-958
-15	-819	-15+(-819)=-834
-21	-585	-21+(-585)=-606
-27	-455	-27+(-455)=-482
-35	-351	-35+(-351)=-386
-39	-315	-39+(-315)=-354
-45	-273	-45+(-273)=-318
-63	-195	-63+(-195)=-258
-65	-189	-65+(-189)=-254
-91	-135	-91+(-135)=-226
-105	-117	-105+(-117)=-222

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-105\"$ and $\"-117\"$ add to $\"-222\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-105\"$ and $\"-117\"$ both multiply to $\"12285\"$ and add to $\"-222\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-222x\"$ with $\"-105x-117x\"$ . Remember, $\"-105\"$ and $\"-117\"$ add to $\"-222\"$ . So this shows us that $\"-105x-117x=-222x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"135x%5E2%2Bhighlight%28-105x-117x%29%2B91\"$ Replace the second term $\"-222x\"$ with $\"-105x-117x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28135x%5E2-105x%29%2B%28-117x%2B91%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"15x%289x-7%29%2B%28-117x%2B91%29\"$ Factor out the GCF $\"15x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"15x%289x-7%29-13%289x-7%29\"$ Factor out $\"13\"$ 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( " $\"%2815x-13%29%289x-7%29\"$ Combine like terms. Or factor out the common term $\"9x-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"135x%5E2-222x%2B91\"$ factors to $\"%2815x-13%29%289x-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"135x%5E2-222x%2B91=%2815x-13%29%289x-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%2815x-13%29%289x-7%29\"$ to get $\"135x%5E2-222x%2B91\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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