SOLUTION: Factoring Polynomials.. m to the second power + 2m - 24

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Question 658085: Factoring Polynomials..
m to the second power + 2m - 24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression m%5E2%2B2m-24, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -24.


Now multiply the first coefficient 1 by the last term -24 to get %281%29%28-24%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


Factors of -24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers -4 and 6 add to 2 (the middle coefficient).


So the two numbers -4 and 6 both multiply to -24 and add to 2


Now replace the middle term 2m with -4m%2B6m. Remember, -4 and 6 add to 2. So this shows us that -4m%2B6m=2m.


m%5E2%2Bhighlight%28-4m%2B6m%29-24 Replace the second term 2m with -4m%2B6m.


%28m%5E2-4m%29%2B%286m-24%29 Group the terms into two pairs.


m%28m-4%29%2B%286m-24%29 Factor out the GCF m from the first group.


m%28m-4%29%2B6%28m-4%29 Factor out 6 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.


%28m%2B6%29%28m-4%29 Combine like terms. Or factor out the common term m-4


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Answer:


So m%5E2%2B2m-24 factors to %28m%2B6%29%28m-4%29.


In other words, m%5E2%2B2m-24=%28m%2B6%29%28m-4%29.


Note: you can check the answer by expanding %28m%2B6%29%28m-4%29 to get m%5E2%2B2m-24 or by graphing the original expression and the answer (the two graphs should be identical).