SOLUTION: Factor by using trial factors. 8x^2y-27xy+9

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Question 615200: Factor by using trial factors. 8x^2y-27xy+9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 8x%5E2y-27xy%2B9, we can see that the first coefficient is 8, the second coefficient is -27, and the last coefficient is 9.


Now multiply the first coefficient 8 by the last coefficient 9 to get %288%29%289%29=72.


Now the question is: what two whole numbers multiply to 72 (the previous product) and add to the second coefficient -27?


To find these two numbers, we need to list all of the factors of 72 (the previous product).


Factors of 72:
1,2,3,4,6,8,9,12,18,24,36,72
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72


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


These factors pair up and multiply to 72.
1*72 = 72
2*36 = 72
3*24 = 72
4*18 = 72
6*12 = 72
8*9 = 72
(-1)*(-72) = 72
(-2)*(-36) = 72
(-3)*(-24) = 72
(-4)*(-18) = 72
(-6)*(-12) = 72
(-8)*(-9) = 72

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -27:


First NumberSecond NumberSum
1721+72=73
2362+36=38
3243+24=27
4184+18=22
6126+12=18
898+9=17
-1-72-1+(-72)=-73
-2-36-2+(-36)=-38
-3-24-3+(-24)=-27
-4-18-4+(-18)=-22
-6-12-6+(-12)=-18
-8-9-8+(-9)=-17



From the table, we can see that the two numbers -3 and -24 add to -27 (the middle coefficient).


So the two numbers -3 and -24 both multiply to 72 and add to -27


Now replace the middle term -27xy with -3xy-24xy. Remember, -3 and -24 add to -27. So this shows us that -3xy-24xy=-27xy.


8x%5E2y%2Bhighlight%28-3xy-24xy%29%2B9 Replace the second term -27xy with -3xy-24xy.


%288x%5E2y-3xy%29%2B%28-24xy%2B9%29 Group the terms into two pairs.


xy%288x-3%29%2B%28-24xy%2B9%29 Factor out the GCF xy from the first group.


xy%288x-3%29-3%288xy-3%29 Factor out 3 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


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Answer:


So 8x%5E2y-27xy%2B9 factors to %28xy-3%29%288x-3%29.


In other words, 8x%5E2y-27xy%2B9=%28xy-3%29%288x-3%29.


Note: you can check the answer by expanding %28xy-3%29%288x-3%29 to get 8x%5E2y-27xy%2B9 or by graphing the original expression and the answer (the two graphs should be identical).