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

Problem

(x−3)3

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 and n=3

2. Set up the expansion using the binomial coefficients (3/0)) (3/1)) (3/2)) and (3/3))

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

3. Calculate the binomial coefficients using the formula (n/k)=(n!)/(k!(n−k)!))

(3/0)=1)

(3/1)=3)

(3/2)=3)

(3/3)=1)

4. Substitute the coefficients and powers back into the expression.

(x−3)3=1*(x3)*(1)+3*(x2)*(−3)+3*(x)*(9)+1*(1)*(−27)

5. Simplify each term to find the final polynomial.

(x−3)3=x3−9*x2+27*x−27

Final Answer

(x−3)3=x3−9*x2+27*x−27

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