(a+b)^3=a^3+b^3

Jasonbradley

2 min read

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Jun 16, 2024

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The Incorrect Expansion: (a + b)³ ≠ a³ + b³

It's a common misconception that the cube of a sum is simply the sum of the cubes. While this might seem intuitive at first glance, it's not true.

Let's break it down:

(a + b)³ represents the product of (a + b) multiplied by itself three times:

(a + b)³ = (a + b) * (a + b) * (a + b)

To correctly expand this expression, we need to apply the distributive property (often called FOIL) multiple times:

1. Expand the first two factors: (a + b) * (a + b) = a² + 2ab + b²

2. Multiply the result by the remaining factor (a + b): (a² + 2ab + b²) * (a + b) = a³ + 3a²b + 3ab² + b³

Therefore, the correct expansion of (a + b)³ is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Key Takeaways:

• The expression (a + b)³ ≠ a³ + b³.
• Expanding the cube of a sum requires applying the distributive property repeatedly.
• The correct expansion results in four terms, not just two.