SOLUTION: What are the factors of n^2+16n-36?

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Question 195818: What are the factors of n^2+16n-36?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression n%5E2%2B16n-36, we can see that the first coefficient is 1, the second coefficient is 16, and the last term is -36.


Now multiply the first coefficient 1 by the last term -36 to get %281%29%28-36%29=-36.


Now the question is: what two whole numbers multiply to -36 (the previous product) and add to the second coefficient 16?


To find these two numbers, we need to list all of the factors of -36 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 16:


First NumberSecond NumberSum
1-361+(-36)=-35
2-182+(-18)=-16
3-123+(-12)=-9
4-94+(-9)=-5
6-66+(-6)=0
-136-1+36=35
-218-2+18=16
-312-3+12=9
-49-4+9=5
-66-6+6=0



From the table, we can see that the two numbers -2 and 18 add to 16 (the middle coefficient).


So the two numbers -2 and 18 both multiply to -36 and add to 16


Now replace the middle term 16n with -2n%2B18n. Remember, -2 and 18 add to 16. So this shows us that -2n%2B18n=16n.


n%5E2%2Bhighlight%28-2n%2B18n%29-36 Replace the second term 16n with -2n%2B18n.


%28n%5E2-2n%29%2B%2818n-36%29 Group the terms into two pairs.


n%28n-2%29%2B%2818n-36%29 Factor out the GCF n from the first group.


n%28n-2%29%2B18%28n-2%29 Factor out 18 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.


%28n%2B18%29%28n-2%29 Combine like terms. Or factor out the common term n-2

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Answer:


So n%5E2%2B16n-36 factors to %28n%2B18%29%28n-2%29.


Note: you can check the answer by FOILing %28n%2B18%29%28n-2%29 to get n%5E2%2B16n-36 or by graphing the original expression and the answer (the two graphs should be identical).