The Expanded Form of (a  b)^3
The expression (a  b)^3 represents the cube of the binomial (a  b). To understand its expanded form, we can use the distributive property and the binomial theorem.
Using the Distributive Property
We can expand (a  b)^3 step by step using the distributive property:

First, square the binomial: (a  b)^2 = (a  b)(a  b) = a^2  ab  ba + b^2 = a^2  2ab + b^2

Now, multiply the result by (a  b): (a  b)^3 = (a^2  2ab + b^2)(a  b) = a^3  2a^2b + ab^2  a^2b + 2ab^2  b^3 = a^3  3a^2b + 3ab^2  b^3
Using the Binomial Theorem
The binomial theorem provides a general formula for expanding any binomial raised to a power. For (a  b)^3, the theorem gives us:
(a  b)^3 = ¹C₀a³(b)⁰ + ¹C₁a²(b)¹ + ¹C₂a¹(b)² + ¹C₃a⁰(b)³
Where ¹Cₓ represents the binomial coefficient, calculated as ¹Cₓ = ¹! / (ₓ! * (¹ₓ)!).
Evaluating the coefficients and simplifying, we get:
(a  b)^3 = a³  3a²b + 3ab²  b³
Summary
The expanded form of (a  b)^3 is:
(a  b)^3 = a³  3a²b + 3ab²  b³
This formula is useful for various algebraic manipulations and problemsolving, particularly when dealing with cubic expressions.