SOLUTION: Could you please help me factor the following polynomial? 4y^2+16y+16 Thank you, Viola

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Question 251756: Could you please help me factor the following polynomial?
4y^2+16y+16
Thank you,
Viola
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
factor the following polynomial?
4y^2+16y+16
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= 4(y^2 + 4y + 4)
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= 4(y+2)^2
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Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

4y%5E2%2B16y%2B16 Start with the given expression.


4%28y%5E2%2B4y%2B4%29 Factor out the GCF 4.


Now let's try to factor the inner expression y%5E2%2B4y%2B4


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Looking at the expression y%5E2%2B4y%2B4, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is 4.


Now multiply the first coefficient 1 by the last term 4 to get %281%29%284%29=4.


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


To find these two numbers, we need to list all of the factors of 4 (the previous product).


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


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


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

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


First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



From the table, we can see that the two numbers 2 and 2 add to 4 (the middle coefficient).


So the two numbers 2 and 2 both multiply to 4 and add to 4


Now replace the middle term 4y with 2y%2B2y. Remember, 2 and 2 add to 4. So this shows us that 2y%2B2y=4y.


y%5E2%2Bhighlight%282y%2B2y%29%2B4 Replace the second term 4y with 2y%2B2y.


%28y%5E2%2B2y%29%2B%282y%2B4%29 Group the terms into two pairs.


y%28y%2B2%29%2B%282y%2B4%29 Factor out the GCF y from the first group.


y%28y%2B2%29%2B2%28y%2B2%29 Factor out 2 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%2B2%29%28y%2B2%29 Combine like terms. Or factor out the common term y%2B2


%28y%2B2%29%5E2 Condense the terms.


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So 4%28y%5E2%2B4y%2B4%29 then factors further to 4%28y%2B2%29%5E2


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Answer:


So 4y%5E2%2B16y%2B16 completely factors to 4%28y%2B2%29%5E2.


In other words, 4y%5E2%2B16y%2B16=4%28y%2B2%29%5E2.


Note: you can check the answer by expanding 4%28y%2B2%29%5E2 to get 4y%5E2%2B16y%2B16 or by graphing the original expression and the answer (the two graphs should be identical).