document.write( "Question 411181: factor x^2-200x-1800 \n" ); document.write( "

Algebra.Com's Answer #289018 by MathLover1(20850) $\"\"$ $\"About$

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

Looking at the expression $\"x%5E2%2B200x%2B1800\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"200\"$ , and the last term is $\"1800\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"1800\"$ to get $\"%281%29%281800%29=1800\"$ .

Now the question is: what two whole numbers multiply to $\"1800\"$ (the previous product) and add to the second coefficient $\"200\"$ ?

To find these two numbers, we need to list all of the factors of $\"1800\"$ (the previous product).

Factors of $\"1800\"$ :

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

-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 $\"1800\"$ .

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
(-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 $\"200\"$ :

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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 $\"200\"$ . So $\"x%5E2%2B200x%2B1800\"$ cannot be factored.

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Answer:

So $\"x%5E2%2B200%2Ax%2B1800\"$ doesn't factor at all (over the rational numbers).

So $\"x%5E2%2B200%2Ax%2B1800\"$ is prime.

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