SOLUTION: How do you factor (x^2+2xy+y^2)-4w^2?

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Question 183267: How do you factor (x^2+2xy+y^2)-4w^2?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%5E2%2B2xy%2By%5E2%29-4w%5E2 Start with the given expression.


First, we need to factor x%5E2%2B2xy%2By%5E2




Looking at x%5E2%2B2xy%2By%5E2 we can see that the first term is x%5E2 and the last term is y%5E2 where the coefficients are 1 and 1 respectively.

Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient 2? Let's list all of the factors of 1:



Factors of 1:
1

-1 ...List the negative factors as well. This will allow us to find all possible combinations

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

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

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



From this list we can see that 1 and 1 add up to 2 and multiply to 1


Now looking at the expression x%5E2%2B2xy%2By%5E2, replace 2xy with xy%2Bxy (notice xy%2Bxy adds up to 2xy. So it is equivalent to 2xy)

x%5E2%2Bhighlight%28xy%2Bxy%29%2By%5E2


Now let's factor x%5E2%2Bxy%2Bxy%2By%5E2 by grouping:


%28x%5E2%2Bxy%29%2B%28xy%2By%5E2%29 Group like terms


x%28x%2By%29%2By%28x%2By%29 Factor out the GCF of x out of the first group. Factor out the GCF of y out of the second group


%28x%2By%29%28x%2By%29 Since we have a common term of x%2By, we can combine like terms


So x%5E2%2Bxy%2Bxy%2By%5E2 factors to %28x%2By%29%28x%2By%29


So this also means that x%5E2%2B2xy%2By%5E2 factors to %28x%2By%29%28x%2By%29 (since x%5E2%2B2xy%2B1y%5E2 is equivalent to x%5E2%2Bxy%2Bxy%2By%5E2)


note: %28x%2By%29%28x%2By%29 is equivalent to %28x%2By%29%5E2 since the term x%2By occurs twice. So x%5E2%2B2xy%2By%5E2 also factors to %28x%2By%29%5E2


So x%5E2%2B2xy%2By%5E2 factors to %28x%2By%29%5E2


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So the original expression %28x%5E2%2B2xy%2By%5E2%29-4w%5E2 becomes %28x%2By%29%5E2-4w%5E2


%28x%2By%29%5E2-%282w%29%5E2 Rewrite 4w%5E2 as %282w%29%5E2


Notice how we now have a difference of squares. Remember, the difference of squares formula is A%5E2-B%5E2=%28A%2BB%29%28A-B%29


In this case, A=x%2By and B=2w


Now plug them in the formula to get:


%28x%2By%29%5E2-%282w%29%5E2=%28x%2By%2B2w%29%28x%2By-2w%29



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Answer:


So this means that %28x%2By%29%5E2-%282w%29%5E2 factors to %28x%2By%2B2w%29%28x%2By-2w%29


Equivalently, this means that %28x%5E2%2B2xy%2By%5E2%29-4w%5E2 factors to %28x%2By%2B2w%29%28x%2By-2w%29