SOLUTION: 72x^3-11x^2-380x

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Question 422690: 72x^3-11x^2-380x
Found 2 solutions by jim_thompson5910, richard1234:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
I'm assuming you want to factor this.




Start with the given expression.


Factor out the GCF .


Now let's try to factor the inner expression


---------------------------------------------------------------


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
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
-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


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*(-27360) = -27360
2*(-13680) = -27360
3*(-9120) = -27360
4*(-6840) = -27360
5*(-5472) = -27360
6*(-4560) = -27360
8*(-3420) = -27360
9*(-3040) = -27360
10*(-2736) = -27360
12*(-2280) = -27360
15*(-1824) = -27360
16*(-1710) = -27360
18*(-1520) = -27360
19*(-1440) = -27360
20*(-1368) = -27360
24*(-1140) = -27360
30*(-912) = -27360
32*(-855) = -27360
36*(-760) = -27360
38*(-720) = -27360
40*(-684) = -27360
45*(-608) = -27360
48*(-570) = -27360
57*(-480) = -27360
60*(-456) = -27360
72*(-380) = -27360
76*(-360) = -27360
80*(-342) = -27360
90*(-304) = -27360
95*(-288) = -27360
96*(-285) = -27360
114*(-240) = -27360
120*(-228) = -27360
144*(-190) = -27360
152*(-180) = -27360
160*(-171) = -27360
(-1)*(27360) = -27360
(-2)*(13680) = -27360
(-3)*(9120) = -27360
(-4)*(6840) = -27360
(-5)*(5472) = -27360
(-6)*(4560) = -27360
(-8)*(3420) = -27360
(-9)*(3040) = -27360
(-10)*(2736) = -27360
(-12)*(2280) = -27360
(-15)*(1824) = -27360
(-16)*(1710) = -27360
(-18)*(1520) = -27360
(-19)*(1440) = -27360
(-20)*(1368) = -27360
(-24)*(1140) = -27360
(-30)*(912) = -27360
(-32)*(855) = -27360
(-36)*(760) = -27360
(-38)*(720) = -27360
(-40)*(684) = -27360
(-45)*(608) = -27360
(-48)*(570) = -27360
(-57)*(480) = -27360
(-60)*(456) = -27360
(-72)*(380) = -27360
(-76)*(360) = -27360
(-80)*(342) = -27360
(-90)*(304) = -27360
(-95)*(288) = -27360
(-96)*(285) = -27360
(-114)*(240) = -27360
(-120)*(228) = -27360
(-144)*(190) = -27360
(-152)*(180) = -27360
(-160)*(171) = -27360

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :


First NumberSecond NumberSum
1-273601+(-27360)=-27359
2-136802+(-13680)=-13678
3-91203+(-9120)=-9117
4-68404+(-6840)=-6836
5-54725+(-5472)=-5467
6-45606+(-4560)=-4554
8-34208+(-3420)=-3412
9-30409+(-3040)=-3031
10-273610+(-2736)=-2726
12-228012+(-2280)=-2268
15-182415+(-1824)=-1809
16-171016+(-1710)=-1694
18-152018+(-1520)=-1502
19-144019+(-1440)=-1421
20-136820+(-1368)=-1348
24-114024+(-1140)=-1116
30-91230+(-912)=-882
32-85532+(-855)=-823
36-76036+(-760)=-724
38-72038+(-720)=-682
40-68440+(-684)=-644
45-60845+(-608)=-563
48-57048+(-570)=-522
57-48057+(-480)=-423
60-45660+(-456)=-396
72-38072+(-380)=-308
76-36076+(-360)=-284
80-34280+(-342)=-262
90-30490+(-304)=-214
95-28895+(-288)=-193
96-28596+(-285)=-189
114-240114+(-240)=-126
120-228120+(-228)=-108
144-190144+(-190)=-46
152-180152+(-180)=-28
160-171160+(-171)=-11
-127360-1+27360=27359
-213680-2+13680=13678
-39120-3+9120=9117
-46840-4+6840=6836
-55472-5+5472=5467
-64560-6+4560=4554
-83420-8+3420=3412
-93040-9+3040=3031
-102736-10+2736=2726
-122280-12+2280=2268
-151824-15+1824=1809
-161710-16+1710=1694
-181520-18+1520=1502
-191440-19+1440=1421
-201368-20+1368=1348
-241140-24+1140=1116
-30912-30+912=882
-32855-32+855=823
-36760-36+760=724
-38720-38+720=682
-40684-40+684=644
-45608-45+608=563
-48570-48+570=522
-57480-57+480=423
-60456-60+456=396
-72380-72+380=308
-76360-76+360=284
-80342-80+342=262
-90304-90+304=214
-95288-95+288=193
-96285-96+285=189
-114240-114+240=126
-120228-120+228=108
-144190-144+190=46
-152180-152+180=28
-160171-160+171=11



From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out 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.


Combine like terms. Or factor out the common term


--------------------------------------------------


So then factors further to


===============================================================


Answer:


So completely factors to .


In other words, .


Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim
Answer by richard1234(7193)   (Show Source): You can put this solution on YOUR website!
An easier way to factor this is somewhat "cheating" (not really) but the best way to factor a strange expression like this is to find the zeros of the function. We can factor x out:

. Either x = 0 or . By the quadratic formula,

. I won't guide you through all the arithmetic but this simplifies to x = 19/8 and x = -20/9. Hence our polynomial is in the form . It helps to have integer coefficients so we can bring out an 8 to the first term and a 9 to the second term to obtain the factorization .
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