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 :
1,2,3,4,5,6,8,9,10,12,15,18,20,24,25,30,36,40,45,50,60,72,75,90,100,120,150,180,200,225,300,360,450,600,900,1800
-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-18,-20,-24,-25,-30,-36,-40,-45,-50,-60,-72,-75,-90,-100,-120,-150,-180,-200,-225,-300,-360,-450,-600,-900,-1800
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*1800 = 1800
2*900 = 1800
3*600 = 1800
4*450 = 1800
5*360 = 1800
6*300 = 1800
8*225 = 1800
9*200 = 1800
10*180 = 1800
12*150 = 1800
15*120 = 1800
18*100 = 1800
20*90 = 1800
24*75 = 1800
25*72 = 1800
30*60 = 1800
36*50 = 1800
40*45 = 1800
(-1)*(-1800) = 1800
(-2)*(-900) = 1800
(-3)*(-600) = 1800
(-4)*(-450) = 1800
(-5)*(-360) = 1800
(-6)*(-300) = 1800
(-8)*(-225) = 1800
(-9)*(-200) = 1800
(-10)*(-180) = 1800
(-12)*(-150) = 1800
(-15)*(-120) = 1800
(-18)*(-100) = 1800
(-20)*(-90) = 1800
(-24)*(-75) = 1800
(-25)*(-72) = 1800
(-30)*(-60) = 1800
(-36)*(-50) = 1800
(-40)*(-45) = 1800
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 1800 | 1+1800=1801 | | 2 | 900 | 2+900=902 | | 3 | 600 | 3+600=603 | | 4 | 450 | 4+450=454 | | 5 | 360 | 5+360=365 | | 6 | 300 | 6+300=306 | | 8 | 225 | 8+225=233 | | 9 | 200 | 9+200=209 | | 10 | 180 | 10+180=190 | | 12 | 150 | 12+150=162 | | 15 | 120 | 15+120=135 | | 18 | 100 | 18+100=118 | | 20 | 90 | 20+90=110 | | 24 | 75 | 24+75=99 | | 25 | 72 | 25+72=97 | | 30 | 60 | 30+60=90 | | 36 | 50 | 36+50=86 | | 40 | 45 | 40+45=85 | | -1 | -1800 | -1+(-1800)=-1801 | | -2 | -900 | -2+(-900)=-902 | | -3 | -600 | -3+(-600)=-603 | | -4 | -450 | -4+(-450)=-454 | | -5 | -360 | -5+(-360)=-365 | | -6 | -300 | -6+(-300)=-306 | | -8 | -225 | -8+(-225)=-233 | | -9 | -200 | -9+(-200)=-209 | | -10 | -180 | -10+(-180)=-190 | | -12 | -150 | -12+(-150)=-162 | | -15 | -120 | -15+(-120)=-135 | | -18 | -100 | -18+(-100)=-118 | | -20 | -90 | -20+(-90)=-110 | | -24 | -75 | -24+(-75)=-99 | | -25 | -72 | -25+(-72)=-97 | | -30 | -60 | -30+(-60)=-90 | | -36 | -50 | -36+(-50)=-86 | | -40 | -45 | -40+(-45)=-85 |
From the table, we can see that there are no pairs of numbers which add to . So cannot be factored.
===============================================================
Answer:
So  doesn't factor at all (over the rational numbers).
So is prime.
|
