Expand Using the Binomial Theorem (x+y)^4

Problem

(x+y)^4

Solution

1. Identify the parameters of the Binomial Theorem (a+b)^n where a=x b=y and n=4

2. Apply the formula for the binomial expansion, which states (a+b)^n=(∑_k=0^n)((n/k))*a^(n−k)*b^k)

3. Write out the terms for k=0,1,2,3,4

(4/0)x^4*y^0+(4/1)x^3*y^1+(4/2)x^2*y^2+(4/3)x^1*y^3+(4/4)x^0*y^4)))))

4. Calculate the binomial coefficients (n/k)) using Pascal's triangle or the formula (n!)/(k!(n−k)!)

(4/0)=1)

(4/1)=4)

(4/2)=6)

(4/3)=4)

(4/4)=1)

5. Substitute the coefficients back into the expansion and simplify the powers:

1*x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+1*y^4

Final Answer

(x+y)^4=x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4

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