SOLUTION: 3h^2+6ht+3t

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Question 738423: 3h^2+6ht+3t
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If the expression is 3h%5E2%2B6ht%2B3t%5E2, then you can do this to factor



3h%5E2%2B6ht%2B3t%5E2 Start with the given expression.


3%28h%5E2%2B2ht%2Bt%5E2%29 Factor out the GCF 3.


Now let's try to factor the inner expression h%5E2%2B2ht%2Bt%5E2


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Looking at the expression h%5E2%2B2ht%2Bt%5E2, we can see that the first coefficient is 1, the second coefficient is 2, and the last coefficient is 1.


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


Now the question is: what two whole numbers multiply to 1 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of 1 (the previous product).


Factors of 1:
1
-1


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2



From the table, we can see that the two numbers 1 and 1 add to 2 (the middle coefficient).


So the two numbers 1 and 1 both multiply to 1 and add to 2


Now replace the middle term 2ht with ht%2Bht. Remember, 1 and 1 add to 2. So this shows us that ht%2Bht=2ht.


h%5E2%2Bhighlight%28ht%2Bht%29%2Bt%5E2 Replace the second term 2ht with ht%2Bht.


%28h%5E2%2Bht%29%2B%28ht%2Bt%5E2%29 Group the terms into two pairs.


h%28h%2Bt%29%2B%28ht%2Bt%5E2%29 Factor out the GCF h from the first group.


h%28h%2Bt%29%2Bt%28h%2Bt%29 Factor out t 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.


%28h%2Bt%29%28h%2Bt%29 Combine like terms. Or factor out the common term h%2Bt


%28h%2Bt%29%5E2 Condense the terms.


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So 3%28h%5E2%2B2ht%2Bt%5E2%29 then factors further to 3%28h%2Bt%29%5E2


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Answer:


So 3h%5E2%2B6ht%2B3t%5E2 completely factors to 3%28h%2Bt%29%5E2.


In other words, 3h%5E2%2B6ht%2B3t%5E2=3%28h%2Bt%29%5E2.


Note: you can check the answer by expanding 3%28h%2Bt%29%5E2 to get 3h%5E2%2B6ht%2B3t%5E2 or by graphing the original expression and the answer (the two graphs should be identical).