SOLUTION: Factor x^2-xy-56y^2

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Question 582002: Factor x^2-xy-56y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2-xy-56y%5E2, we can see that the first coefficient is 1, the second coefficient is -1, and the last coefficient is -56.


Now multiply the first coefficient 1 by the last coefficient -56 to get %281%29%28-56%29=-56.


Now the question is: what two whole numbers multiply to -56 (the previous product) and add to the second coefficient -1?


To find these two numbers, we need to list all of the factors of -56 (the previous product).


Factors of -56:
1,2,4,7,8,14,28,56
-1,-2,-4,-7,-8,-14,-28,-56


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


These factors pair up and multiply to -56.
1*(-56) = -56
2*(-28) = -56
4*(-14) = -56
7*(-8) = -56
(-1)*(56) = -56
(-2)*(28) = -56
(-4)*(14) = -56
(-7)*(8) = -56

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:


First NumberSecond NumberSum
1-561+(-56)=-55
2-282+(-28)=-26
4-144+(-14)=-10
7-87+(-8)=-1
-156-1+56=55
-228-2+28=26
-414-4+14=10
-78-7+8=1



From the table, we can see that the two numbers 7 and -8 add to -1 (the middle coefficient).


So the two numbers 7 and -8 both multiply to -56 and add to -1


Now replace the middle term -1xy with 7xy-8xy. Remember, 7 and -8 add to -1. So this shows us that 7xy-8xy=-1xy.


x%5E2%2Bhighlight%287xy-8xy%29-56y%5E2 Replace the second term -1xy with 7xy-8xy.


%28x%5E2%2B7xy%29%2B%28-8xy-56y%5E2%29 Group the terms into two pairs.


x%28x%2B7y%29%2B%28-8xy-56y%5E2%29 Factor out the GCF x from the first group.


x%28x%2B7y%29-8y%28x%2B7y%29 Factor out -8y from the second group.


%28x-8y%29%28x%2B7y%29 Factor out x%2B7y


So x%5E2-xy-56y%5E2 completely factors to %28x-8y%29%28x%2B7y%29