Expand Using the Binomial Theorem (x+5)^3

Problem

(x+5)3

Solution

1. Identify the components of the binomial expression (a+b)n where a=x b=5 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 out the terms of the expansion using the coefficients from the 3rd row of Pascal's Triangle (1, 3, 3, 1$).

(x+5)3=(3/0)x3*5+(3/1)x2*5+(3/2)x1*5+(3/3)x0*5))))

4. Calculate the binomial coefficients and the powers of 5.

(x+5)3=1⋅x3⋅1+3⋅x2⋅5+3⋅x⋅25+1⋅1⋅125

5. Simplify each term by multiplying the constants.

(x+5)3=x3+15*x2+75*x+125

Final Answer

(x+5)3=x3+15*x2+75*x+125

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