SOLUTION: factor x^2-200x-1800

Question 411181: factor x^2-200x-1800
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

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.

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
It's fairly tedious listing all positive/negative factors of -1800 so I'll solve using the quadratic formula. The roots of are

. Therefore the polynomial can be factored as . However, it's still a pretty ugly expression, and the polynomial cannot be factored over the integers.
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