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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression 28m%5E2-41mn%2B15n%5E2, we can see that the first coefficient is 28, the second coefficient is -41, and the last coefficient is 15.


Now multiply the first coefficient 28 by the last coefficient 15 to get %2828%29%2815%29=420.


Now the question is: what two whole numbers multiply to 420 (the previous product) and add to the second coefficient -41?


To find these two numbers, we need to list all of the factors of 420 (the previous product).


Factors of 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
-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 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
(-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 -41:


First NumberSecond NumberSum
14201+420=421
22102+210=212
31403+140=143
41054+105=109
5845+84=89
6706+70=76
7607+60=67
104210+42=52
123512+35=47
143014+30=44
152815+28=43
202120+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 -20 and -21 add to -41 (the middle coefficient).


So the two numbers -20 and -21 both multiply to 420 and add to -41


Now replace the middle term -41mn with -20mn-21mn. Remember, -20 and -21 add to -41. So this shows us that -20mn-21mn=-41mn.


28m%5E2%2Bhighlight%28-20mn-21mn%29%2B15n%5E2 Replace the second term -41mn with -20mn-21mn.


%2828m%5E2-20mn%29%2B%28-21mn%2B15n%5E2%29 Group the terms into two pairs.


4m%287m-5n%29%2B%28-21mn%2B15n%5E2%29 Factor out the GCF 4m from the first group.


4m%287m-5n%29-3n%287m-5n%29 Factor out -3n from the second group.


%284m-3n%29%287m-5n%29 Factor out 7m-5n
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So 28m%5E2-41mn%2B15n%5E2 completely factors to %284m-3n%29%287m-5n%29