Expand Using the Binomial Theorem (x-y)^5

Problem

(x−y)5

Solution

1. Identify the parameters for the Binomial Theorem formula (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk) Here, a=x b=−y and n=5

2. Write the expansion using the binomial coefficients (5/k)) for k=0,1,2,3,4,5

(5/0)x5*(−y)0+(5/1)x4*(−y)1+(5/2)x3*(−y)2+(5/3)x2*(−y)3+(5/4)x1*(−y)4+(5/5)x0*(−y)5))))))

3. Calculate the binomial coefficients using Pascal's triangle or the formula (n/k)=(n!)/(k!(n−k)!)) The coefficients for n=5 are 1, 5, 10, 10, 5, 1$.

1*x(1)5+5*x4*(−y)+10*x(y2)3+10*x2*(−y3)+5*x(y4)+1*(−y5)

4. Simplify each term by applying the signs resulting from the powers of −y

x5−5*x4*y+10*x3*y2−10*x2*y3+5*x*y4−y5

Final Answer

(x−y)5=x5−5*x4*y+10*x3*y2−10*x2*y3+5*x*y4−y5

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