Expand Using the Binomial Theorem (2x-y)^5

Problem

(2*x−y)5

Solution

1. Identify the components of the binomial (a+b)n where a=2*x b=−y and n=5

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

3. Write out the summation terms for k=0 to k=5

(5/0)(2*x)5*(−y)0+(5/1)(2*x)4*(−y)1+(5/2)(2*x)3*(−y)2+(5/3)(2*x)2*(−y)3+(5/4)(2*x)1*(−y)4+(5/5)(2*x)0*(−y)5))))))

4. Calculate the binomial coefficients (n/k)) using the values 1, 5, 10, 10, 5, 1$:

1*(2*x)5*(1)+5*(2*x)4*(−y)+10*(2*x)3*(−y)2+10*(2*x)2*(−y)3+5*(2*x)*(−y)4+1*(−y)5

5. Evaluate the powers of each term:

1*(32*x5)*(1)+5*(16*x4)*(−y)+10*(8*x3)*(y2)+10*(4*x2)*(−y3)+5*(2*x)*(y4)+1*(−y5)

6. Simplify each term by multiplying the coefficients and variables:

32*x5−80*x4*y+80*x3*y2−40*x2*y3+10*x*y4−y5

Final Answer

(2*x−y)5=32*x5−80*x4*y+80*x3*y2−40*x2*y3+10*x*y4−y5

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