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

Problem

(x−y)3

Solution

1. Identify the parameters for the Binomial Theorem formula (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk) where a=x b=−y and n=3

2. Write the expansion by applying the formula for each value of k from 0 to 3

(x−y)3=(3/0)x3*(−y)0+(3/1)x2*(−y)1+(3/2)x1*(−y)2+(3/3)x0*(−y)3))))

3. Calculate the binomial coefficients (n/k)) using the values from Pascal's triangle for n=3 which are 1, 3, 3, 1$.

(3/0)=1)

(3/1)=3)

(3/2)=3)

(3/3)=1)

4. Simplify each term by multiplying the coefficients and handling the signs of the powers of −y

(x−y)3=1*(x3)*(1)+3*(x2)*(−y)+3*(x)*(y2)+1*(1)*(−y3)

(x−y)3=x3−3*x2*y+3*x*y2−y3

Final Answer

(x−y)3=x3−3*x2*y+3*x*y2−y3

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