SOLUTION: Factor. −3y^3 + 15y^2 − 18y

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Question 778421: Factor.
−3y^3 + 15y^2 − 18y
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

-3y%5E3%2B15y%5E2-18y Start with the given expression.


-3y%28y%5E2-5y%2B6%29 Factor out the GCF -3y.


Now let's try to factor the inner expression y%5E2-5y%2B6


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Looking at the expression y%5E2-5y%2B6, we can see that the first coefficient is 1, the second coefficient is -5, and the last term is 6.


Now multiply the first coefficient 1 by the last term 6 to get %281%29%286%29=6.


Now the question is: what two whole numbers multiply to 6 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of 6 (the previous product).


Factors of 6:
1,2,3,6
-1,-2,-3,-6


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
161+6=7
232+3=5
-1-6-1+(-6)=-7
-2-3-2+(-3)=-5



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


So the two numbers -2 and -3 both multiply to 6 and add to -5


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


y%5E2%2Bhighlight%28-2y-3y%29%2B6 Replace the second term -5y with -2y-3y.


%28y%5E2-2y%29%2B%28-3y%2B6%29 Group the terms into two pairs.


y%28y-2%29%2B%28-3y%2B6%29 Factor out the GCF y from the first group.


y%28y-2%29-3%28y-2%29 Factor out 3 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-3%29%28y-2%29 Combine like terms. Or factor out the common term y-2


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So -3y%28y%5E2-5y%2B6%29 then factors further to -3y%28y-3%29%28y-2%29


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Answer:


So -3y%5E3%2B15y%5E2-18y completely factors to -3y%28y-3%29%28y-2%29.


In other words, -3y%5E3%2B15y%5E2-18y=-3y%28y-3%29%28y-2%29.


Note: you can check the answer by expanding -3y%28y-3%29%28y-2%29 to get -3y%5E3%2B15y%5E2-18y or by graphing the original expression and the answer (the two graphs should be identical).