(a-b)^3 Expanded Form

Jasonbradley

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Jun 16, 2024

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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:

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

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! * (¹-ₓ)!).

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 problem-solving, particularly when dealing with cubic expressions.