The given formula is the difference of cubes. It is expressed as:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Explanation of the Formula:

1. Expression Breakdown:

   • $a^3 - b^3$ represents the difference of the cubes of $a$ and $b$.
   • This can be factored into two components:
     1. A linear factor: $a - b$
     2. A quadratic factor: $a^2 + ab + b^2$

2. Verification by Expansion: To verify, expand the right-hand side: $(a - b)(a^2 + ab + b^2) = a(a^2 + ab + b^2) - b(a^2 + ab + b^2)$

   • Expanding the terms: $= a^3 + a^2b + ab^2 - (b^3 + a^2b + ab^2)$
   • Simplify: $= a^3 - b^3$

3. Application:

   • This formula is commonly used in algebra to simplify expressions or solve equations involving cubic terms.

Example:

Factorize $x^3 - 8$:

1. Recognize that $x^3 - 8 = x^3 - 2^3$.
2. Apply the difference of cubes formula: $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$

Would you like a deeper explanation, an example with values, or further details? Let me know!

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Related Questions:

1. Can you apply this formula to $27y^3 - 1$?
2. What is the sum of cubes formula? How is it different?
3. How does this formula connect to polynomial division?
4. Can you derive the quadratic term $a^2 + ab + b^2$ geometrically?
5. Is there a similar formula for the difference of fourth powers?

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Tip:

Always check for common factors before applying the difference of cubes formula for efficient simplification!