SOLUTION: Show all stages of factoring separately. Factor completely: 3x2 – 2x – 8

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Question 110731: Show all stages of factoring separately.
Factor completely:
3x2 – 2x – 8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 3x%5E2-2x-8, we can see that the first coefficient is 3, the second coefficient is -2, and the last term is -8.



Now multiply the first coefficient 3 by the last term -8 to get %283%29%28-8%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 -2x with 4x-6x. Remember, 4 and -6 add to -2. So this shows us that 4x-6x=-2x.



3x%5E2%2Bhighlight%284x-6x%29-8 Replace the second term -2x with 4x-6x.



%283x%5E2%2B4x%29%2B%28-6x-8%29 Group the terms into two pairs.



x%283x%2B4%29%2B%28-6x-8%29 Factor out the GCF x from the first group.



x%283x%2B4%29-2%283x%2B4%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.



%28x-2%29%283x%2B4%29 Combine like terms. Or factor out the common term 3x%2B4



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Answer:



So 3%2Ax%5E2-2%2Ax-8 factors to %28x-2%29%283x%2B4%29.



In other words, 3%2Ax%5E2-2%2Ax-8=%28x-2%29%283x%2B4%29.



Note: you can check the answer by expanding %28x-2%29%283x%2B4%29 to get 3%2Ax%5E2-2%2Ax-8 or by graphing the original expression and the answer (the two graphs should be identical).