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:

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

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.