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

Problem

(x+3)^3

Solution

1. Identify the components of the binomial (a+b)^n where a=x b=3 and n=3

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 terms for k=0,1,2,3

(3/0)x^3*3^0+(3/1)x^2*3^1+(3/2)x^1*3^2+(3/3)x^0*3^3))))

4. Calculate the binomial coefficients (n/k)) using the values $1, 3, 3, 1$ from Pascal's Triangle:

1⋅x^3⋅1+3⋅x^2⋅3+3⋅x⋅9+1⋅1⋅27

5. Simplify each term by performing the multiplication:

x^3+9*x^2+27*x+27

Final Answer

(x+3)^3=x^3+9*x^2+27*x+27

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