SOLUTION: FACTOR TRIANOMIAL COMPLETELY t^2+16t+64 =?^2

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Question 207568This question is from textbook beginning algebra
: FACTOR TRIANOMIAL COMPLETELY
t^2+16t+64 =?^2 This question is from textbook beginning algebra

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


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Looking at the expression t%5E2%2B16t%2B64, we can see that the first coefficient is 1, the second coefficient is 16, and the last term is 64.


Now multiply the first coefficient 1 by the last term 64 to get %281%29%2864%29=64.


Now the question is: what two whole numbers multiply to 64 (the previous product) and add to the second coefficient 16?


To find these two numbers, we need to list all of the factors of 64 (the previous product).


Factors of 64:
1,2,4,8,16,32,64
-1,-2,-4,-8,-16,-32,-64


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


These factors pair up and multiply to 64.
1*64 = 64
2*32 = 64
4*16 = 64
8*8 = 64
(-1)*(-64) = 64
(-2)*(-32) = 64
(-4)*(-16) = 64
(-8)*(-8) = 64

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 16:


First NumberSecond NumberSum
1641+64=65
2322+32=34
4164+16=20
888+8=16
-1-64-1+(-64)=-65
-2-32-2+(-32)=-34
-4-16-4+(-16)=-20
-8-8-8+(-8)=-16



From the table, we can see that the two numbers 8 and 8 add to 16 (the middle coefficient).


So the two numbers 8 and 8 both multiply to 64 and add to 16


Now replace the middle term 16t with 8t%2B8t. Remember, 8 and 8 add to 16. So this shows us that 8t%2B8t=16t.


t%5E2%2Bhighlight%288t%2B8t%29%2B64 Replace the second term 16t with 8t%2B8t.


%28t%5E2%2B8t%29%2B%288t%2B64%29 Group the terms into two pairs.


t%28t%2B8%29%2B%288t%2B64%29 Factor out the GCF t from the first group.


t%28t%2B8%29%2B8%28t%2B8%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.


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


%28t%2B8%29%5E2 Condense the terms.


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Answer:


So t%5E2%2B16t%2B64 factors to %28t%2B8%29%5E2.


In other words, t%5E2%2B16t%2B64=%28t%2B8%29%5E2.


Note: you can check the answer by expanding %28t%2B8%29%5E2 to get t%5E2%2B16t%2B64 or by graphing the original expression and the answer (the two graphs should be identical).

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