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

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: 72x^3-11x^2-380x       Log On


   

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




72x%5E3-11x%5E2-380x Start with the given expression.


x%2872x%5E2-11x-380%29 Factor out the GCF x.


Now let's try to factor the inner expression 72x%5E2-11x-380


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


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.


Now multiply the first coefficient 72 by the last term -380 to get %2872%29%28-380%29=-27360.


Now the question is: what two whole numbers multiply to -27360 (the previous product) and add to the second coefficient -11?


To find these two numbers, we need to list all of the factors of -27360 (the previous product).


Factors of -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
-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 -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
(-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 -11:


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 160 and -171 add to -11 (the middle coefficient).


So the two numbers 160 and -171 both multiply to -27360 and add to -11


Now replace the middle term -11x with 160x-171x. Remember, 160 and -171 add to -11. So this shows us that 160x-171x=-11x.


72x%5E2%2Bhighlight%28160x-171x%29-380 Replace the second term -11x with 160x-171x.


%2872x%5E2%2B160x%29%2B%28-171x-380%29 Group the terms into two pairs.


8x%289x%2B20%29%2B%28-171x-380%29 Factor out the GCF 8x from the first group.


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.


%288x-19%29%289x%2B20%29 Combine like terms. Or factor out the common term 9x%2B20


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


So x%2872x%5E2-11x-380%29 then factors further to x%288x-19%29%289x%2B20%29


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


Answer:


So 72x%5E3-11x%5E2-380x completely factors to x%288x-19%29%289x%2B20%29.


In other words, 72x%5E3-11x%5E2-380x=x%288x-19%29%289x%2B20%29.


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).


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) About Me  (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:

x%2872x%5E2+-+11x+-+380%29+=+0. Either x = 0 or 72x%5E2+-+11x+-+380+=+0. By the quadratic formula,

x+=+%2811+%2B-+sqrt%28121+%2B+4%2872%29%28380%29%29%29%2F144. 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 x%288x+-+19%29%289x+%2B+20%29.