SOLUTION: y^3-12y^2+32y factor trinomial completely

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Question 627003: y^3-12y^2+32y factor trinomial completely
Answer by jim_thompson5910(35256) About Me  (Show Source):
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y%5E3-12y%5E2%2B32y Start with the given expression.


y%28y%5E2-12y%2B32%29 Factor out the GCF y.


Now let's try to factor the inner expression y%5E2-12y%2B32


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Looking at the expression y%5E2-12y%2B32, we can see that the first coefficient is 1, the second coefficient is -12, and the last term is 32.


Now multiply the first coefficient 1 by the last term 32 to get %281%29%2832%29=32.


Now the question is: what two whole numbers multiply to 32 (the previous product) and add to the second coefficient -12?


To find these two numbers, we need to list all of the factors of 32 (the previous product).


Factors of 32:
1,2,4,8,16,32
-1,-2,-4,-8,-16,-32


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


These factors pair up and multiply to 32.
1*32 = 32
2*16 = 32
4*8 = 32
(-1)*(-32) = 32
(-2)*(-16) = 32
(-4)*(-8) = 32

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


First NumberSecond NumberSum
1321+32=33
2162+16=18
484+8=12
-1-32-1+(-32)=-33
-2-16-2+(-16)=-18
-4-8-4+(-8)=-12



From the table, we can see that the two numbers -4 and -8 add to -12 (the middle coefficient).


So the two numbers -4 and -8 both multiply to 32 and add to -12


Now replace the middle term -12y with -4y-8y. Remember, -4 and -8 add to -12. So this shows us that -4y-8y=-12y.


y%5E2%2Bhighlight%28-4y-8y%29%2B32 Replace the second term -12y with -4y-8y.


%28y%5E2-4y%29%2B%28-8y%2B32%29 Group the terms into two pairs.


y%28y-4%29%2B%28-8y%2B32%29 Factor out the GCF y from the first group.


y%28y-4%29-8%28y-4%29 Factor out 8 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-8%29%28y-4%29 Combine like terms. Or factor out the common term y-4


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So y%28y%5E2-12y%2B32%29 then factors further to y%28y-8%29%28y-4%29


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Answer:


So y%5E3-12y%5E2%2B32y completely factors to y%28y-8%29%28y-4%29.


In other words, y%5E3-12y%5E2%2B32y=y%28y-8%29%28y-4%29.


Note: you can check the answer by expanding y%28y-8%29%28y-4%29 to get y%5E3-12y%5E2%2B32y or by graphing the original expression and the answer (the two graphs should be identical).

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