(a+b)3
If you meant (a+b)^3 then
a^3 + 3a^2b + 3ab^2 + b^3
(a+b)^3 =
(a+b)^2(a+b) = a^2 + 2ab + b^2 +
a^2b + 2ab^2 + b^3,
Combine like-terms:
a^3 + 3a^2b + 3ab^2 + b^3.
To simplify the expression (a+b)^3, we need to expand it using the binomial expansion formula. In general, the formula for expanding (a+b)^n, where n is a positive integer, is:
(a+b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n-1)ab^(n-1) + C(n,n)b^n
where C(n,r) represents the binomial coefficient, given by:
C(n,r) = n! / (r!(n-r)!)
In the case of (a+b)^3, we have n = 3. Plugging this into the formula, we get:
(a+b)^3 = C(3,0)a^3 + C(3,1)a^2b + C(3,2)ab^2 + C(3,3)b^3
Evaluating the binomial coefficients, we have:
C(3,0) = 1
C(3,1) = 3
C(3,2) = 3
C(3,3) = 1
Substituting these back into the formula, we get:
(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3
Simplifying further, we have:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Therefore, the expanded form of (a+b)^3 is a^3 + 3a^2b + 3ab^2 + b^3.