Expand Using the Binomial Theorem (2x+3)^3

Problem

(2*x+3)3

Solution

1. Identify the components of the binomial (a+b)n where a=2*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)*bk)

3. Write the expansion by substituting the values into each term of the sum.

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

4. Calculate the binomial coefficients (3/0)=1) (3/1)=3) (3/2)=3) and (3/3)=1)

5. Simplify each term by evaluating the powers and multiplying the coefficients.

1*(8*x3)*(1)+3*(4*x2)*(3)+3*(2*x)*(9)+1*(1)*(27)

6. Combine the results to find the final polynomial.

8*x3+36*x2+54*x+27

Final Answer

(2*x+3)3=8*x3+36*x2+54*x+27

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