SOLUTION: please help me solve this problem: (x+y)^4

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Question 198002: please help me solve this problem: (x+y)^4
Found 2 solutions by checkley77, jim_thompson5910:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
(x+y)^4=(x+y)^2*(x+y)^2
(x+y)^2
x+y
x+y Multiply.
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x^2+2xy+y^2
x^2+2xy+y^2 Multiply
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x^4+2x^3y+x^2y^2+2x^3y+4x^2y^2+2xy^3+x^2y^2+2xy^3+y^4
x^4+4x^3y+6x^2y^2+4xy^3+y^4 ans.

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


%28x%2By%29%5E4 Start with the given expression

To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle:
1   

1   1   

1   2   1   

1   3   3   1   

1   4   6   4   1   




Looking at the row that starts with 1,4, etc, we can see that this row has the numbers:

1, 4, 6, 4, and 1

These numbers will be the coefficients of our expansion. So to expand %28x%2By%29%5E4, simply follow this procedure:
Write the first coefficient. Multiply that coefficient with the first binomial term x and then the second binomial term y. Repeat this until all of the coefficients have been written.

Once that has been done, add up the terms like this:


Notice how the coefficients are in front of each term.



However, we're not done yet.


Looking at the first term 1%28x%29%28y%29, raise x to the 4th power and raise y to the 0th power.

Looking at the second term 4%28x%29%28y%29 raise x to the 3rd power and raise y to the 1st power.

Continue this until you reach the final term.


Notice how the exponents of x are stepping down and the exponents of y are stepping up.


So the fully expanded expression should now look like this:





Distribute the exponents


1%28x%5E4%29%2B4%28x%5E3y%29%2B6%28x%5E2y%5E2%29%2B4%28xy%5E3%29%2B1%28y%5E4%29 Multiply


x%5E4%2B4x%5E3y%2B6x%5E2y%5E2%2B4xy%5E3%2By%5E4 Multiply the terms with their coefficients


So %28x%2By%29%5E4 expands and simplifies to x%5E4%2B4x%5E3y%2B6x%5E2y%5E2%2B4xy%5E3%2By%5E4.


In other words, %28x%2By%29%5E4=x%5E4%2B4x%5E3y%2B6x%5E2y%5E2%2B4xy%5E3%2By%5E4