SOLUTION: Factor the trinomial x^2-x-56

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Question 190255: Factor the trinomial
x^2-x-56
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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


Now multiply the first coefficient 1 by the last term -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)
2*(-28)
4*(-14)
7*(-8)
(-1)*(56)
(-2)*(28)
(-4)*(14)
(-7)*(8)

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 -1x with 7x-8x. Remember, 7 and -8 add to -1. So this shows us that 7x-8x=-1x.


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


%28x%5E2%2B7x%29%2B%28-8x-56%29 Group the terms into two pairs.


x%28x%2B7%29%2B%28-8x-56%29 Factor out the GCF x from the first group.


x%28x%2B7%29-8%28x%2B7%29 Factor out 8 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.


%28x-8%29%28x%2B7%29 Combine like terms. Or factor out the common term x%2B7

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Answer:


So x%5E2-x-56 factors to %28x-8%29%28x%2B7%29.


Note: you can check the answer by FOILing %28x-8%29%28x%2B7%29 to get x%5E2-x-56 or by graphing the original expression and the answer (the two graphs should be identical).