Understanding the Whole Cube of a - b

What is the whole cube of a - b?

When dealing with algebraic expressions, the whole cube of a binomial expression is a fundamental concept. Specifically, the whole cube of the binomial a - b can be expanded using the binomial formula. This formula states that:

a - b^3  a^3 - 3a^2b   3ab^2 - b^3

To break this down further:

a - b is a binomial. (a^3 - 3a^2b 3ab^2 - b^3) is the expanded form of the whole cube of a - b.

Expanding the Expression

Let's explore how this expression is derived step-by-step:

a - b^3  a^3 - 3a^2b   3ab^2 - b^3

This can be further broken down:

a - b^3  a^3 - 3a^2b   3ab^2 - b^3  a^3 - 3a^2b   3ab^2 - b^3  aa^2 - 3ab   3b^2 - b^3

a - b^3  a^3 - 3a^2b   3ab^2 - b^3  [a - b] × [a - b] × [a - b]

By multiplying the binomial [a - b] with itself three times, we get the expanded form:

[a - b] × [a - b] × [a - b]  a^3 - 3a^2b   3ab^2 - b^3

Verifying the Identity

To ensure the accuracy of this identity, you can also verify it by performing the multiplication:

[a - b] × [a - b] a^2 - 2ab b^2[a^2 - 2ab b^2] × (a - b) a^3 - 2a^2b ab^2 - a^2b 2ab^2 - b^3Simplifying, we get: a^3 - 3a^2b 3ab^2 - b^3

Algebraic Identity Representation

The identity can also be represented for convenience in an expanded form:

a - b^3  a^3 - 3a^2b   3ab^2 - b^3

This can be further simplified and represented as:

a - b^3  a^3 - b^3 - 3ab(a - b)

Therefore, the whole cube of a - b can be succinctly written as:

a - b^3  a^3 - 3a^2b   3ab^2 - b^3

Understanding the whole cube of a binomial expression is crucial in algebra, and remembering the identity can greatly simplify complex problems.