SOLUTION: factor the following trinomial completely: 3x^2+12x+12

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Question 530360: factor the following trinomial completely: 3x^2+12x+12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E2%2B12x%2B12 Start with the given expression.


3%28x%5E2%2B4x%2B4%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2%2B4x%2B4


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Looking at the expression x%5E2%2B4x%2B4, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is 4.


Now multiply the first coefficient 1 by the last term 4 to get %281%29%284%29=4.


Now the question is: what two whole numbers multiply to 4 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of 4 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



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


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


Now replace the middle term 4x with 2x%2B2x. Remember, 2 and 2 add to 4. So this shows us that 2x%2B2x=4x.


x%5E2%2Bhighlight%282x%2B2x%29%2B4 Replace the second term 4x with 2x%2B2x.


%28x%5E2%2B2x%29%2B%282x%2B4%29 Group the terms into two pairs.


x%28x%2B2%29%2B%282x%2B4%29 Factor out the GCF x from the first group.


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


%28x%2B2%29%5E2 Condense the terms.


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So 3%28x%5E2%2B4x%2B4%29 then factors further to 3%28x%2B2%29%5E2


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Answer:


So 3x%5E2%2B12x%2B12 completely factors to 3%28x%2B2%29%5E2.


In other words, 3x%5E2%2B12x%2B12=3%28x%2B2%29%5E2.


Note: you can check the answer by expanding 3%28x%2B2%29%5E2 to get 3x%5E2%2B12x%2B12 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

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Jim