SOLUTION: Could you help me with this problem about factoring polynomials completely 2x^2+5x-12 Thank you for your time and help!

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Question 640407: Could you help me with this problem about factoring polynomials completely
2x^2+5x-12
Thank you for your time and help!

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 2x%5E2%2B5x-12, we can see that the first coefficient is 2, the second coefficient is 5, and the last term is -12.


Now multiply the first coefficient 2 by the last term -12 to get %282%29%28-12%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) = -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 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 5x with -3x%2B8x. Remember, -3 and 8 add to 5. So this shows us that -3x%2B8x=5x.


2x%5E2%2Bhighlight%28-3x%2B8x%29-12 Replace the second term 5x with -3x%2B8x.


%282x%5E2-3x%29%2B%288x-12%29 Group the terms into two pairs.


x%282x-3%29%2B%288x-12%29 Factor out the GCF x from the first group.


x%282x-3%29%2B4%282x-3%29 Factor out 4 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%2B4%29%282x-3%29 Combine like terms. Or factor out the common term 2x-3


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Answer:


So 2x%5E2%2B5x-12 factors to %28x%2B4%29%282x-3%29.


In other words, 2x%5E2%2B5x-12=%28x%2B4%29%282x-3%29.


Note: you can check the answer by expanding %28x%2B4%29%282x-3%29 to get 2x%5E2%2B5x-12 or by graphing the original expression and the answer (the two graphs should be identical).