Expand Using the Binomial Theorem (x-3y)^7

Problem

(x−3*y)7

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=−3*y and n=7

2. Set up the summation by writing out each of the 8 terms using the binomial coefficients (7/k)) for k=0,1,2,3,4,5,6,7

3. Calculate the binomial coefficients: (7/0)=1) (7/1)=7) (7/2)=21) (7/3)=35) (7/4)=35) (7/5)=21) (7/6)=7) and (7/7)=1)

4. Substitute the values into the expansion:

(7/0)x7*(−3*y)0+(7/1)x6*(−3*y)1+(7/2)x5*(−3*y)2+(7/3)x4*(−3*y)3+(7/4)x3*(−3*y)4+(7/5)x2*(−3*y)5+(7/6)x1*(−3*y)6+(7/7)x0*(−3*y)7))))))))

5. Simplify each term by raising −3 to the appropriate power and multiplying by the binomial coefficient.

6. Combine the results into the final polynomial expression.

Final Answer

(x−3*y)7=x7−21*x6*y+189*x5*y2−945*x4*y3+2835*x3*y4−5103*x2*y5+5103*x*y6−2187*y7

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