document.write( "Question 422690: 72x^3-11x^2-380x
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Algebra.Com's Answer #294986 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor this.\r
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\n" ); document.write( "\n" ); document.write( " $\"72x%5E3-11x%5E2-380x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%2872x%5E2-11x-380%29\"$ Factor out the GCF $\"x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"72x%5E2-11x-380\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"72x%5E2-11x-380\"$ , we can see that the first coefficient is $\"72\"$ , the second coefficient is $\"-11\"$ , and the last term is $\"-380\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"72\"$ by the last term $\"-380\"$ to get $\"%2872%29%28-380%29=-27360\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-27360\"$ (the previous product) and add to the second coefficient $\"-11\"$ ?\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 $\"-27360\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-27360\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,9,10,12,15,16,18,19,20,24,30,32,36,38,40,45,48,57,60,72,76,80,90,95,96,114,120,144,152,160,171,180,190,228,240,285,288,304,342,360,380,456,480,570,608,684,720,760,855,912,1140,1368,1440,1520,1710,1824,2280,2736,3040,3420,4560,5472,6840,9120,13680,27360\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-16,-18,-19,-20,-24,-30,-32,-36,-38,-40,-45,-48,-57,-60,-72,-76,-80,-90,-95,-96,-114,-120,-144,-152,-160,-171,-180,-190,-228,-240,-285,-288,-304,-342,-360,-380,-456,-480,-570,-608,-684,-720,-760,-855,-912,-1140,-1368,-1440,-1520,-1710,-1824,-2280,-2736,-3040,-3420,-4560,-5472,-6840,-9120,-13680,-27360\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 $\"-27360\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-27360) = -27360
\n" ); document.write( "2*(-13680) = -27360
\n" ); document.write( "3*(-9120) = -27360
\n" ); document.write( "4*(-6840) = -27360
\n" ); document.write( "5*(-5472) = -27360
\n" ); document.write( "6*(-4560) = -27360
\n" ); document.write( "8*(-3420) = -27360
\n" ); document.write( "9*(-3040) = -27360
\n" ); document.write( "10*(-2736) = -27360
\n" ); document.write( "12*(-2280) = -27360
\n" ); document.write( "15*(-1824) = -27360
\n" ); document.write( "16*(-1710) = -27360
\n" ); document.write( "18*(-1520) = -27360
\n" ); document.write( "19*(-1440) = -27360
\n" ); document.write( "20*(-1368) = -27360
\n" ); document.write( "24*(-1140) = -27360
\n" ); document.write( "30*(-912) = -27360
\n" ); document.write( "32*(-855) = -27360
\n" ); document.write( "36*(-760) = -27360
\n" ); document.write( "38*(-720) = -27360
\n" ); document.write( "40*(-684) = -27360
\n" ); document.write( "45*(-608) = -27360
\n" ); document.write( "48*(-570) = -27360
\n" ); document.write( "57*(-480) = -27360
\n" ); document.write( "60*(-456) = -27360
\n" ); document.write( "72*(-380) = -27360
\n" ); document.write( "76*(-360) = -27360
\n" ); document.write( "80*(-342) = -27360
\n" ); document.write( "90*(-304) = -27360
\n" ); document.write( "95*(-288) = -27360
\n" ); document.write( "96*(-285) = -27360
\n" ); document.write( "114*(-240) = -27360
\n" ); document.write( "120*(-228) = -27360
\n" ); document.write( "144*(-190) = -27360
\n" ); document.write( "152*(-180) = -27360
\n" ); document.write( "160*(-171) = -27360
\n" ); document.write( "(-1)*(27360) = -27360
\n" ); document.write( "(-2)*(13680) = -27360
\n" ); document.write( "(-3)*(9120) = -27360
\n" ); document.write( "(-4)*(6840) = -27360
\n" ); document.write( "(-5)*(5472) = -27360
\n" ); document.write( "(-6)*(4560) = -27360
\n" ); document.write( "(-8)*(3420) = -27360
\n" ); document.write( "(-9)*(3040) = -27360
\n" ); document.write( "(-10)*(2736) = -27360
\n" ); document.write( "(-12)*(2280) = -27360
\n" ); document.write( "(-15)*(1824) = -27360
\n" ); document.write( "(-16)*(1710) = -27360
\n" ); document.write( "(-18)*(1520) = -27360
\n" ); document.write( "(-19)*(1440) = -27360
\n" ); document.write( "(-20)*(1368) = -27360
\n" ); document.write( "(-24)*(1140) = -27360
\n" ); document.write( "(-30)*(912) = -27360
\n" ); document.write( "(-32)*(855) = -27360
\n" ); document.write( "(-36)*(760) = -27360
\n" ); document.write( "(-38)*(720) = -27360
\n" ); document.write( "(-40)*(684) = -27360
\n" ); document.write( "(-45)*(608) = -27360
\n" ); document.write( "(-48)*(570) = -27360
\n" ); document.write( "(-57)*(480) = -27360
\n" ); document.write( "(-60)*(456) = -27360
\n" ); document.write( "(-72)*(380) = -27360
\n" ); document.write( "(-76)*(360) = -27360
\n" ); document.write( "(-80)*(342) = -27360
\n" ); document.write( "(-90)*(304) = -27360
\n" ); document.write( "(-95)*(288) = -27360
\n" ); document.write( "(-96)*(285) = -27360
\n" ); document.write( "(-114)*(240) = -27360
\n" ); document.write( "(-120)*(228) = -27360
\n" ); document.write( "(-144)*(190) = -27360
\n" ); document.write( "(-152)*(180) = -27360
\n" ); document.write( "(-160)*(171) = -27360\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 $\"-11\"$ :\r
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First Number	Second Number	Sum
1	-27360	1+(-27360)=-27359
2	-13680	2+(-13680)=-13678
3	-9120	3+(-9120)=-9117
4	-6840	4+(-6840)=-6836
5	-5472	5+(-5472)=-5467
6	-4560	6+(-4560)=-4554
8	-3420	8+(-3420)=-3412
9	-3040	9+(-3040)=-3031
10	-2736	10+(-2736)=-2726
12	-2280	12+(-2280)=-2268
15	-1824	15+(-1824)=-1809
16	-1710	16+(-1710)=-1694
18	-1520	18+(-1520)=-1502
19	-1440	19+(-1440)=-1421
20	-1368	20+(-1368)=-1348
24	-1140	24+(-1140)=-1116
30	-912	30+(-912)=-882
32	-855	32+(-855)=-823
36	-760	36+(-760)=-724
38	-720	38+(-720)=-682
40	-684	40+(-684)=-644
45	-608	45+(-608)=-563
48	-570	48+(-570)=-522
57	-480	57+(-480)=-423
60	-456	60+(-456)=-396
72	-380	72+(-380)=-308
76	-360	76+(-360)=-284
80	-342	80+(-342)=-262
90	-304	90+(-304)=-214
95	-288	95+(-288)=-193
96	-285	96+(-285)=-189
114	-240	114+(-240)=-126
120	-228	120+(-228)=-108
144	-190	144+(-190)=-46
152	-180	152+(-180)=-28
160	-171	160+(-171)=-11
-1	27360	-1+27360=27359
-2	13680	-2+13680=13678
-3	9120	-3+9120=9117
-4	6840	-4+6840=6836
-5	5472	-5+5472=5467
-6	4560	-6+4560=4554
-8	3420	-8+3420=3412
-9	3040	-9+3040=3031
-10	2736	-10+2736=2726
-12	2280	-12+2280=2268
-15	1824	-15+1824=1809
-16	1710	-16+1710=1694
-18	1520	-18+1520=1502
-19	1440	-19+1440=1421
-20	1368	-20+1368=1348
-24	1140	-24+1140=1116
-30	912	-30+912=882
-32	855	-32+855=823
-36	760	-36+760=724
-38	720	-38+720=682
-40	684	-40+684=644
-45	608	-45+608=563
-48	570	-48+570=522
-57	480	-57+480=423
-60	456	-60+456=396
-72	380	-72+380=308
-76	360	-76+360=284
-80	342	-80+342=262
-90	304	-90+304=214
-95	288	-95+288=193
-96	285	-96+285=189
-114	240	-114+240=126
-120	228	-120+228=108
-144	190	-144+190=46
-152	180	-152+180=28
-160	171	-160+171=11

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"160\"$ and $\"-171\"$ add to $\"-11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"160\"$ and $\"-171\"$ both multiply to $\"-27360\"$ and add to $\"-11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-11x\"$ with $\"160x-171x\"$ . Remember, $\"160\"$ and $\"-171\"$ add to $\"-11\"$ . So this shows us that $\"160x-171x=-11x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"72x%5E2%2Bhighlight%28160x-171x%29-380\"$ Replace the second term $\"-11x\"$ with $\"160x-171x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2872x%5E2%2B160x%29%2B%28-171x-380%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"8x%289x%2B20%29%2B%28-171x-380%29\"$ Factor out the GCF $\"8x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"8x%289x%2B20%29-19%289x%2B20%29\"$ Factor out $\"19\"$ 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( " $\"%288x-19%29%289x%2B20%29\"$ Combine like terms. Or factor out the common term $\"9x%2B20\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%2872x%5E2-11x-380%29\"$ then factors further to $\"x%288x-19%29%289x%2B20%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"72x%5E3-11x%5E2-380x\"$ completely factors to $\"x%288x-19%29%289x%2B20%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"72x%5E3-11x%5E2-380x=x%288x-19%29%289x%2B20%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"x%288x-19%29%289x%2B20%29\"$ to get $\"72x%5E3-11x%5E2-380x\"$ 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( "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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