SOLUTION: Factor the given polynomial 12z^2 – 5z – 2

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Question 154222: Factor the given polynomial
12z^2 – 5z – 2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 12z%5E2-5z-2, we can see that the first coefficient is 12, the second coefficient is -5, and the last term is -2.


Now multiply the first coefficient 12 by the last term -2 to get %2812%29%28-2%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient -5?


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 -5:


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 3 and -8 add to -5 (the middle coefficient).


So the two numbers 3 and -8 both multiply to -24 and add to -5


Now replace the middle term -5z with 3z-8z. Remember, 3 and -8 add to -5. So this shows us that 3z-8z=-5z.


12z%5E2%2Bhighlight%283z-8z%29-2 Replace the second term -5z with 3z-8z.


%2812z%5E2%2B3z%29%2B%28-8z-2%29 Group the terms into two pairs.


3z%284z%2B1%29%2B%28-8z-2%29 Factor out the GCF 3z from the first group.


3z%284z%2B1%29-2%284z%2B1%29 Factor out 2 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.


%283z-2%29%284z%2B1%29 Combine like terms. Or factor out the common term 4z%2B1

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Answer:


So 12z%5E2-5z-2 factors to %283z-2%29%284z%2B1%29.


Note: you can check the answer by FOILing %283z-2%29%284z%2B1%29 to get 12z%5E2-5z-2 or by graphing the original expression and the answer (the two graphs should be identical).