SOLUTION: 3x squared minus 6x minus 24.............im being asked to factor completely

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Question 318721: 3x squared minus 6x minus 24.............im being asked to factor completely
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E2-6x-24 Start with the given expression.


3%28x%5E2-2x-8%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2-2x-8


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Looking at the expression x%5E2-2x-8, we can see that the first coefficient is 1, the second coefficient is -2, and the last term is -8.


Now multiply the first coefficient 1 by the last term -8 to get %281%29%28-8%29=-8.


Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of -8 (the previous product).


Factors of -8:
1,2,4,8
-1,-2,-4,-8


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


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

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


First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2



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


So the two numbers 2 and -4 both multiply to -8 and add to -2


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


x%5E2%2Bhighlight%282x-4x%29-8 Replace the second term -2x with 2x-4x.


%28x%5E2%2B2x%29%2B%28-4x-8%29 Group the terms into two pairs.


x%28x%2B2%29%2B%28-4x-8%29 Factor out the GCF x from the first group.


x%28x%2B2%29-4%28x%2B2%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-4%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2


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So 3%28x%5E2-2x-8%29 then factors further to 3%28x-4%29%28x%2B2%29


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Answer:


So 3x%5E2-6x-24 completely factors to 3%28x-4%29%28x%2B2%29.


In other words, 3x%5E2-6x-24=3%28x-4%29%28x%2B2%29.


Note: you can check the answer by expanding 3%28x-4%29%28x%2B2%29 to get 3x%5E2-6x-24 or by graphing the original expression and the answer (the two graphs should be identical).