SOLUTION: I need to know how to factor this Trinomial. y^3-15y^2+54y

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Question 631225: I need to know how to factor this Trinomial.
y^3-15y^2+54y
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y%5E3-15y%5E2%2B54y Start with the given expression.


y%28y%5E2-15y%2B54%29 Factor out the GCF y.


Now let's try to factor the inner expression y%5E2-15y%2B54


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Looking at the expression y%5E2-15y%2B54, we can see that the first coefficient is 1, the second coefficient is -15, and the last term is 54.


Now multiply the first coefficient 1 by the last term 54 to get %281%29%2854%29=54.


Now the question is: what two whole numbers multiply to 54 (the previous product) and add to the second coefficient -15?


To find these two numbers, we need to list all of the factors of 54 (the previous product).


Factors of 54:
1,2,3,6,9,18,27,54
-1,-2,-3,-6,-9,-18,-27,-54


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


These factors pair up and multiply to 54.
1*54 = 54
2*27 = 54
3*18 = 54
6*9 = 54
(-1)*(-54) = 54
(-2)*(-27) = 54
(-3)*(-18) = 54
(-6)*(-9) = 54

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -15:


First NumberSecond NumberSum
1541+54=55
2272+27=29
3183+18=21
696+9=15
-1-54-1+(-54)=-55
-2-27-2+(-27)=-29
-3-18-3+(-18)=-21
-6-9-6+(-9)=-15



From the table, we can see that the two numbers -6 and -9 add to -15 (the middle coefficient).


So the two numbers -6 and -9 both multiply to 54 and add to -15


Now replace the middle term -15y with -6y-9y. Remember, -6 and -9 add to -15. So this shows us that -6y-9y=-15y.


y%5E2%2Bhighlight%28-6y-9y%29%2B54 Replace the second term -15y with -6y-9y.


%28y%5E2-6y%29%2B%28-9y%2B54%29 Group the terms into two pairs.


y%28y-6%29%2B%28-9y%2B54%29 Factor out the GCF y from the first group.


y%28y-6%29-9%28y-6%29 Factor out 9 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-9%29%28y-6%29 Combine like terms. Or factor out the common term y-6


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So y%28y%5E2-15y%2B54%29 then factors further to y%28y-9%29%28y-6%29


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Answer:


So y%5E3-15y%5E2%2B54y completely factors to y%28y-9%29%28y-6%29.


In other words, y%5E3-15y%5E2%2B54y=y%28y-9%29%28y-6%29.


Note: you can check the answer by expanding y%28y-9%29%28y-6%29 to get y%5E3-15y%5E2%2B54y or by graphing the original expression and the answer (the two graphs should be identical).

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Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


First take a y out of all three terms. That leaves you with a quadratic trinomial factor in y. Just reverse the FOIL process. Hint: -6 plus -9 is -15, -6 times -9 is 54.

John

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