SOLUTION: I need help factoring y^3-14y^2+45y

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Question 635185: I need help factoring y^3-14y^2+45y
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y%5E3-14y%5E2%2B45y Start with the given expression.


y%28y%5E2-14y%2B45%29 Factor out the GCF y.


Now let's try to factor the inner expression y%5E2-14y%2B45


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Looking at the expression y%5E2-14y%2B45, we can see that the first coefficient is 1, the second coefficient is -14, and the last term is 45.


Now multiply the first coefficient 1 by the last term 45 to get %281%29%2845%29=45.


Now the question is: what two whole numbers multiply to 45 (the previous product) and add to the second coefficient -14?


To find these two numbers, we need to list all of the factors of 45 (the previous product).


Factors of 45:
1,3,5,9,15,45
-1,-3,-5,-9,-15,-45


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


These factors pair up and multiply to 45.
1*45 = 45
3*15 = 45
5*9 = 45
(-1)*(-45) = 45
(-3)*(-15) = 45
(-5)*(-9) = 45

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -14:


First NumberSecond NumberSum
1451+45=46
3153+15=18
595+9=14
-1-45-1+(-45)=-46
-3-15-3+(-15)=-18
-5-9-5+(-9)=-14



From the table, we can see that the two numbers -5 and -9 add to -14 (the middle coefficient).


So the two numbers -5 and -9 both multiply to 45 and add to -14


Now replace the middle term -14y with -5y-9y. Remember, -5 and -9 add to -14. So this shows us that -5y-9y=-14y.


y%5E2%2Bhighlight%28-5y-9y%29%2B45 Replace the second term -14y with -5y-9y.


%28y%5E2-5y%29%2B%28-9y%2B45%29 Group the terms into two pairs.


y%28y-5%29%2B%28-9y%2B45%29 Factor out the GCF y from the first group.


y%28y-5%29-9%28y-5%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-5%29 Combine like terms. Or factor out the common term y-5


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So y%28y%5E2-14y%2B45%29 then factors further to y%28y-9%29%28y-5%29


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Answer:


So y%5E3-14y%5E2%2B45y completely factors to y%28y-9%29%28y-5%29.


In other words, y%5E3-14y%5E2%2B45y=y%28y-9%29%28y-5%29.


Note: you can check the answer by expanding y%28y-9%29%28y-5%29 to get y%5E3-14y%5E2%2B45y or by graphing the original expression and the answer (the two graphs should be identical).