SOLUTION: Factor the trinomial completely 12x^3-15x^2-18x

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Question 714872: Factor the trinomial completely
12x^3-15x^2-18x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

12x%5E3-15x%5E2-18x Start with the given expression.


3x%284x%5E2-5x-6%29 Factor out the GCF 3x.


Now let's try to factor the inner expression 4x%5E2-5x-6


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Looking at the expression 4x%5E2-5x-6, we can see that the first coefficient is 4, the second coefficient is -5, and the last term is -6.


Now multiply the first coefficient 4 by the last term -6 to get %284%29%28-6%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-8x. Remember, 3 and -8 add to -5. So this shows us that 3x-8x=-5x.


4x%5E2%2Bhighlight%283x-8x%29-6 Replace the second term -5x with 3x-8x.


%284x%5E2%2B3x%29%2B%28-8x-6%29 Group the terms into two pairs.


x%284x%2B3%29%2B%28-8x-6%29 Factor out the GCF x from the first group.


x%284x%2B3%29-2%284x%2B3%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%284x%2B3%29 Combine like terms. Or factor out the common term 4x%2B3


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So 3x%284x%5E2-5x-6%29 then factors further to 3x%28x-2%29%284x%2B3%29


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Answer:


So 12x%5E3-15x%5E2-18x completely factors to 3x%28x-2%29%284x%2B3%29.


In other words, 12x%5E3-15x%5E2-18x=3x%28x-2%29%284x%2B3%29.


Note: you can check the answer by expanding 3x%28x-2%29%284x%2B3%29 to get 12x%5E3-15x%5E2-18x or by graphing the original expression and the answer (the two graphs should be identical).