SOLUTION: z^2-10z-24

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Question 183248: z^2-10z-24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that you want to factor this.




Looking at the expression z%5E2-10z-24, we can see that the first coefficient is 1, the second coefficient is -10, 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 -10?


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)
2*(-12)
3*(-8)
4*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -10:


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 2 and -12 add to -10 (the middle coefficient).


So the two numbers 2 and -12 both multiply to -24 and add to -10


Now replace the middle term -10z with 2z-12z. Remember, 2 and -12 add to -10. So this shows us that 2z-12z=-10z.


z%5E2%2Bhighlight%282z-12z%29-24 Replace the second term -10z with 2z-12z.


%28z%5E2%2B2z%29%2B%28-12z-24%29 Group the terms into two pairs.


z%28z%2B2%29%2B%28-12z-24%29 Factor out the GCF z from the first group.


z%28z%2B2%29-12%28z%2B2%29 Factor out 12 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.


%28z-12%29%28z%2B2%29 Combine like terms. Or factor out the common term z%2B2

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Answer:


So z%5E2-10z-24 factors to %28z-12%29%28z%2B2%29.


Note: you can check the answer by FOILing %28z-12%29%28z%2B2%29 to get z%5E2-10z-24 or by graphing the original expression and the answer (the two graphs should be identical).