Expand Using the Binomial Theorem (a+b)^3

Problem

(a+b)3

Solution

1. Identify the power n and the terms of the binomial. Here, n=3 the first term is a and the second term is b

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 using the binomial coefficients for n=3 which are (3/0)) (3/1)) (3/2)) and (3/3))

(a+b)3=(3/0)a3*b0+(3/1)a2*b1+(3/2)a1*b2+(3/3)a0*b3))))

4. Calculate the coefficients using the formula (n/k)=(n!)/(k!(n−k)!)) or Pascal's Triangle.

(3/0)=1)

(3/1)=3)

(3/2)=3)

(3/3)=1)

5. Substitute and simplify the expression by combining the coefficients with the variables.

(a+b)3=1*a3+3*a2*b+3*a*b2+1*b3

Final Answer

(a+b)3=a3+3*a2*b+3*a*b2+b3

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