SOLUTION: please help me factor the trinomial {{{21y^3-36y^2-12y}}}

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Question 703574: please help me factor the trinomial 21y%5E3-36y%5E2-12y
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

21y%5E3-36y%5E2-12y Start with the given expression.


3y%287y%5E2-12y-4%29 Factor out the GCF 3y.


Now let's try to factor the inner expression 7y%5E2-12y-4


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Looking at the expression 7y%5E2-12y-4, we can see that the first coefficient is 7, the second coefficient is -12, and the last term is -4.


Now multiply the first coefficient 7 by the last term -4 to get %287%29%28-4%29=-28.


Now the question is: what two whole numbers multiply to -28 (the previous product) and add to the second coefficient -12?


To find these two numbers, we need to list all of the factors of -28 (the previous product).


Factors of -28:
1,2,4,7,14,28
-1,-2,-4,-7,-14,-28


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


These factors pair up and multiply to -28.
1*(-28) = -28
2*(-14) = -28
4*(-7) = -28
(-1)*(28) = -28
(-2)*(14) = -28
(-4)*(7) = -28

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -12:


First NumberSecond NumberSum
1-281+(-28)=-27
2-142+(-14)=-12
4-74+(-7)=-3
-128-1+28=27
-214-2+14=12
-47-4+7=3



From the table, we can see that the two numbers 2 and -14 add to -12 (the middle coefficient).


So the two numbers 2 and -14 both multiply to -28 and add to -12


Now replace the middle term -12y with 2y-14y. Remember, 2 and -14 add to -12. So this shows us that 2y-14y=-12y.


7y%5E2%2Bhighlight%282y-14y%29-4 Replace the second term -12y with 2y-14y.


%287y%5E2%2B2y%29%2B%28-14y-4%29 Group the terms into two pairs.


y%287y%2B2%29%2B%28-14y-4%29 Factor out the GCF y from the first group.


y%287y%2B2%29-2%287y%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.


%28y-2%29%287y%2B2%29 Combine like terms. Or factor out the common term 7y%2B2


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So 3y%287y%5E2-12y-4%29 then factors further to 3y%28y-2%29%287y%2B2%29


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Answer:


So 21y%5E3-36y%5E2-12y completely factors to 3y%28y-2%29%287y%2B2%29.


In other words, 21y%5E3-36y%5E2-12y=3y%28y-2%29%287y%2B2%29.


Note: you can check the answer by expanding 3y%28y-2%29%287y%2B2%29 to get 21y%5E3-36y%5E2-12y or by graphing the original expression and the answer (the two graphs should be identical).