Expand Using the Binomial Theorem (x+y)^3

Problem

(x+y)3

Solution

1. Identify the parameters of the Binomial Theorem, where the expression is (a+b)n with a=x b=y and n=3

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

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

(3/0)x3*y0+(3/1)x2*y1+(3/2)x1*y2+(3/3)x0*y3))))

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

(3/0)=1)

(3/1)=3)

(3/2)=3)

(3/3)=1)

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

1*x3+3*x2*y+3*x*y2+1*y3

Final Answer

(x+y)3=x3+3*x2*y+3*x*y2+y3

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