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)*b^k)

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*x^5)*(1)+5*(16*x^4)*(−y)+10*(8*x^3)*(y^2)+10*(4*x^2)*(−y^3)+5*(2*x)*(y^4)+1*(−y^5)

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

32*x^5−80*x^4*y+80*x^3*y^2−40*x^2*y^3+10*x*y^4−y^5

Final Answer

(2*x−y)^5=32*x^5−80*x^4*y+80*x^3*y^2−40*x^2*y^3+10*x*y^4−y^5

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