SOLUTION: 28m^2-41mn+15n^2 use trial and error to factor trinomial or state that it is prime. I am totally confused

Question 642453: 28m^2-41mn+15n^2 use trial and error to factor trinomial or state that it is prime. I am totally confused
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last coefficient is .

Now multiply the first coefficient by the last coefficient 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 :
1,2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420
-1,-2,-3,-4,-5,-6,-7,-10,-12,-14,-15,-20,-21,-28,-30,-35,-42,-60,-70,-84,-105,-140,-210,-420

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

These factors pair up and multiply to .
1*420 = 420
2*210 = 420
3*140 = 420
4*105 = 420
5*84 = 420
6*70 = 420
7*60 = 420
10*42 = 420
12*35 = 420
14*30 = 420
15*28 = 420
20*21 = 420
(-1)*(-420) = 420
(-2)*(-210) = 420
(-3)*(-140) = 420
(-4)*(-105) = 420
(-5)*(-84) = 420
(-6)*(-70) = 420
(-7)*(-60) = 420
(-10)*(-42) = 420
(-12)*(-35) = 420
(-14)*(-30) = 420
(-15)*(-28) = 420
(-20)*(-21) = 420

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	420	1+420=421
2	210	2+210=212
3	140	3+140=143
4	105	4+105=109
5	84	5+84=89
6	70	6+70=76
7	60	7+60=67
10	42	10+42=52
12	35	12+35=47
14	30	14+30=44
15	28	15+28=43
20	21	20+21=41
-1	-420	-1+(-420)=-421
-2	-210	-2+(-210)=-212
-3	-140	-3+(-140)=-143
-4	-105	-4+(-105)=-109
-5	-84	-5+(-84)=-89
-6	-70	-6+(-70)=-76
-7	-60	-7+(-60)=-67
-10	-42	-10+(-42)=-52
-12	-35	-12+(-35)=-47
-14	-30	-14+(-30)=-44
-15	-28	-15+(-28)=-43
-20	-21	-20+(-21)=-41

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.

Factor out
-------------------------------------------------------

So completely factors to
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