SOLUTION: Factor. Check by Expanding n sqaure+n-42

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Question 551188: Factor. Check by Expanding
n sqaure+n-42
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression n%5E2%2Bn-42, we can see that the first coefficient is 1, the second coefficient is 1, and the last term is -42.


Now multiply the first coefficient 1 by the last term -42 to get %281%29%28-42%29=-42.


Now the question is: what two whole numbers multiply to -42 (the previous product) and add to the second coefficient 1?


To find these two numbers, we need to list all of the factors of -42 (the previous product).


Factors of -42:
1,2,3,6,7,14,21,42
-1,-2,-3,-6,-7,-14,-21,-42


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


These factors pair up and multiply to -42.
1*(-42) = -42
2*(-21) = -42
3*(-14) = -42
6*(-7) = -42
(-1)*(42) = -42
(-2)*(21) = -42
(-3)*(14) = -42
(-6)*(7) = -42

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


First NumberSecond NumberSum
1-421+(-42)=-41
2-212+(-21)=-19
3-143+(-14)=-11
6-76+(-7)=-1
-142-1+42=41
-221-2+21=19
-314-3+14=11
-67-6+7=1



From the table, we can see that the two numbers -6 and 7 add to 1 (the middle coefficient).


So the two numbers -6 and 7 both multiply to -42 and add to 1


Now replace the middle term 1n with -6n%2B7n. Remember, -6 and 7 add to 1. So this shows us that -6n%2B7n=1n.


n%5E2%2Bhighlight%28-6n%2B7n%29-42 Replace the second term 1n with -6n%2B7n.


%28n%5E2-6n%29%2B%287n-42%29 Group the terms into two pairs.


n%28n-6%29%2B%287n-42%29 Factor out the GCF n from the first group.


n%28n-6%29%2B7%28n-6%29 Factor out 7 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%2B7%29%28n-6%29 Combine like terms. Or factor out the common term n-6


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Answer:


So n%5E2%2Bn-42 factors to %28n%2B7%29%28n-6%29.


In other words, n%5E2%2Bn-42=%28n%2B7%29%28n-6%29.


Note: you can check the answer by expanding %28n%2B7%29%28n-6%29 to get n%5E2%2Bn-42 or by graphing the original expression and the answer (the two graphs should be identical).