SOLUTION: x^2+2x+1

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Question 716968: x^2+2x+1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2%2B2x%2B1, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is 1.


Now multiply the first coefficient 1 by the last term 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 2x with x%2Bx. Remember, 1 and 1 add to 2. So this shows us that x%2Bx=2x.


x%5E2%2Bhighlight%28x%2Bx%29%2B1 Replace the second term 2x with x%2Bx.


%28x%5E2%2Bx%29%2B%28x%2B1%29 Group the terms into two pairs.


x%28x%2B1%29%2B%28x%2B1%29 Factor out the GCF x from the first group.


x%28x%2B1%29%2B1%28x%2B1%29 Factor out 1 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.


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


%28x%2B1%29%5E2 Condense the terms.


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Answer:


So x%5E2%2B2x%2B1 factors to %28x%2B1%29%5E2.


In other words, x%5E2%2B2x%2B1=%28x%2B1%29%5E2.


Note: you can check the answer by expanding %28x%2B1%29%5E2 to get x%5E2%2B2x%2B1 or by graphing the original expression and the answer (the two graphs should be identical).