SOLUTION: Factor the trinominal y^4+9y^3+8y^2 (y+8)(y+1)

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Question 652174: Factor the trinominal
y^4+9y^3+8y^2
(y+8)(y+1)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y%5E4%2B9y%5E3%2B8y%5E2 Start with the given expression.


y%5E2%28y%5E2%2B9y%2B8%29 Factor out the GCF y%5E2.


Now let's try to factor the inner expression y%5E2%2B9y%2B8


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Looking at the expression y%5E2%2B9y%2B8, we can see that the first coefficient is 1, the second coefficient is 9, and the last term is 8.


Now multiply the first coefficient 1 by the last term 8 to get %281%29%288%29=8.


Now the question is: what two whole numbers multiply to 8 (the previous product) and add to the second coefficient 9?


To find these two numbers, we need to list all of the factors of 8 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 9:


First NumberSecond NumberSum
181+8=9
242+4=6
-1-8-1+(-8)=-9
-2-4-2+(-4)=-6



From the table, we can see that the two numbers 1 and 8 add to 9 (the middle coefficient).


So the two numbers 1 and 8 both multiply to 8 and add to 9


Now replace the middle term 9y with y%2B8y. Remember, 1 and 8 add to 9. So this shows us that y%2B8y=9y.


y%5E2%2Bhighlight%28y%2B8y%29%2B8 Replace the second term 9y with y%2B8y.


%28y%5E2%2By%29%2B%288y%2B8%29 Group the terms into two pairs.


y%28y%2B1%29%2B%288y%2B8%29 Factor out the GCF y from the first group.


y%28y%2B1%29%2B8%28y%2B1%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%2B8%29%28y%2B1%29 Combine like terms. Or factor out the common term y%2B1


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So y%5E2%28y%5E2%2B9y%2B8%29 then factors further to y%5E2%28y%2B8%29%28y%2B1%29


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Answer:


So y%5E4%2B9y%5E3%2B8y%5E2 completely factors to y%5E2%28y%2B8%29%28y%2B1%29.


In other words, y%5E4%2B9y%5E3%2B8y%5E2=y%5E2%28y%2B8%29%28y%2B1%29.


Note: you can check the answer by expanding y%5E2%28y%2B8%29%28y%2B1%29 to get y%5E4%2B9y%5E3%2B8y%5E2 or by graphing the original expression and the answer (the two graphs should be identical).

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