SOLUTION: reverse foil or grouping 2y^3+24y^2+72y

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Question 299375: reverse foil or grouping
2y^3+24y^2+72y
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

2y%5E3%2B24y%5E2%2B72y Start with the given expression.


2y%28y%5E2%2B12y%2B36%29 Factor out the GCF 2y.


Now let's try to factor the inner expression y%5E2%2B12y%2B36


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Looking at the expression y%5E2%2B12y%2B36, we can see that the first coefficient is 1, the second coefficient is 12, and the last term is 36.


Now multiply the first coefficient 1 by the last term 36 to get %281%29%2836%29=36.


Now the question is: what two whole numbers multiply to 36 (the previous product) and add to the second coefficient 12?


To find these two numbers, we need to list all of the factors of 36 (the previous product).


Factors of 36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to 36.
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
6*6 = 36
(-1)*(-36) = 36
(-2)*(-18) = 36
(-3)*(-12) = 36
(-4)*(-9) = 36
(-6)*(-6) = 36

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


First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From the table, we can see that the two numbers 6 and 6 add to 12 (the middle coefficient).


So the two numbers 6 and 6 both multiply to 36 and add to 12


Now replace the middle term 12y with 6y%2B6y. Remember, 6 and 6 add to 12. So this shows us that 6y%2B6y=12y.


y%5E2%2Bhighlight%286y%2B6y%29%2B36 Replace the second term 12y with 6y%2B6y.


%28y%5E2%2B6y%29%2B%286y%2B36%29 Group the terms into two pairs.


y%28y%2B6%29%2B%286y%2B36%29 Factor out the GCF y from the first group.


y%28y%2B6%29%2B6%28y%2B6%29 Factor out 6 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%2B6%29%28y%2B6%29 Combine like terms. Or factor out the common term y%2B6


%28y%2B6%29%5E2 Condense the terms.


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So 2y%28y%5E2%2B12y%2B36%29 then factors further to 2y%28y%2B6%29%5E2


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Answer:


So 2y%5E3%2B24y%5E2%2B72y completely factors to 2y%28y%2B6%29%5E2.


In other words, 2y%5E3%2B24y%5E2%2B72y=2y%28y%2B6%29%5E2.


Note: you can check the answer by expanding 2y%28y%2B6%29%5E2 to get 2y%5E3%2B24y%5E2%2B72y or by graphing the original expression and the answer (the two graphs should be identical).