SOLUTION: -y^2-4y+5

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Question 225154: -y^2-4y+5
Answer by jim_thompson5910(35256) About Me  (Show Source):
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-y%5E2-4y%2B5 Start with the given expression.


-%28y%5E2%2B4y-5%29 Factor out the GCF -1.


Now let's try to factor the inner expression y%5E2%2B4y-5


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Looking at the expression y%5E2%2B4y-5, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is -5.


Now multiply the first coefficient 1 by the last term -5 to get %281%29%28-5%29=-5.


Now the question is: what two whole numbers multiply to -5 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of -5 (the previous product).


Factors of -5:
1,5
-1,-5


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


These factors pair up and multiply to -5.
1*(-5) = -5
(-1)*(5) = -5

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
1-51+(-5)=-4
-15-1+5=4



From the table, we can see that the two numbers -1 and 5 add to 4 (the middle coefficient).


So the two numbers -1 and 5 both multiply to -5 and add to 4


Now replace the middle term 4y with -y%2B5y. Remember, -1 and 5 add to 4. So this shows us that -y%2B5y=4y.


y%5E2%2Bhighlight%28-y%2B5y%29-5 Replace the second term 4y with -y%2B5y.


%28y%5E2-y%29%2B%285y-5%29 Group the terms into two pairs.


y%28y-1%29%2B%285y-5%29 Factor out the GCF y from the first group.


y%28y-1%29%2B5%28y-1%29 Factor out 5 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%2B5%29%28y-1%29 Combine like terms. Or factor out the common term y-1


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So -%28y%5E2%2B4y-5%29 then factors further to -%28y%2B5%29%28y-1%29


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Answer:


So -y%5E2-4y%2B5 completely factors to -%28y%2B5%29%28y-1%29.


In other words, -y%5E2-4y%2B5=-%28y%2B5%29%28y-1%29.


Note: you can check the answer by expanding -%28y%2B5%29%28y-1%29 to get -y%5E2-4y%2B5 or by graphing the original expression and the answer (the two graphs should be identical).

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