SOLUTION: Could you help me with this problem about factoring polynomials completely 12x^3+8x^2-20x Thank you for your time and help!

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Question 640411: Could you help me with this problem about factoring polynomials completely
12x^3+8x^2-20x
Thank you for your time and help!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

12x%5E3%2B8x%5E2-20x Start with the given expression.


4x%283x%5E2%2B2x-5%29 Factor out the GCF 4x.


Now let's try to factor the inner expression 3x%5E2%2B2x-5


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Looking at the expression 3x%5E2%2B2x-5, we can see that the first coefficient is 3, the second coefficient is 2, and the last term is -5.


Now multiply the first coefficient 3 by the last term -5 to get %283%29%28-5%29=-15.


Now the question is: what two whole numbers multiply to -15 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -15 (the previous product).


Factors of -15:
1,3,5,15
-1,-3,-5,-15


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


These factors pair up and multiply to -15.
1*(-15) = -15
3*(-5) = -15
(-1)*(15) = -15
(-3)*(5) = -15

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


First NumberSecond NumberSum
1-151+(-15)=-14
3-53+(-5)=-2
-115-1+15=14
-35-3+5=2



From the table, we can see that the two numbers -3 and 5 add to 2 (the middle coefficient).


So the two numbers -3 and 5 both multiply to -15 and add to 2


Now replace the middle term 2x with -3x%2B5x. Remember, -3 and 5 add to 2. So this shows us that -3x%2B5x=2x.


3x%5E2%2Bhighlight%28-3x%2B5x%29-5 Replace the second term 2x with -3x%2B5x.


%283x%5E2-3x%29%2B%285x-5%29 Group the terms into two pairs.


3x%28x-1%29%2B%285x-5%29 Factor out the GCF 3x from the first group.


3x%28x-1%29%2B5%28x-1%29 Factor out 5 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.


%283x%2B5%29%28x-1%29 Combine like terms. Or factor out the common term x-1


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So 4x%283x%5E2%2B2x-5%29 then factors further to 4x%283x%2B5%29%28x-1%29


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Answer:


So 12x%5E3%2B8x%5E2-20x completely factors to 4x%283x%2B5%29%28x-1%29.


In other words, 12x%5E3%2B8x%5E2-20x=4x%283x%2B5%29%28x-1%29.


Note: you can check the answer by expanding 4x%283x%2B5%29%28x-1%29 to get 12x%5E3%2B8x%5E2-20x or by graphing the original expression and the answer (the two graphs should be identical).