What is the factoring formula for the sum of cubes?

The sum of cubes factoring formula is given by ( a³ + b³ = (a + b)(a² - ab + b²) ). This formula allows us to express the sum of two cubic terms as a product of a binomial and a trinomial.

To understand why this is the correct formula, we can verify it using algebraic expansion. If we take the right side, ( (a + b)(a² - ab + b²) ), and expand it, we distribute ( (a + b) ) across the trinomial:

1. ( a \cdot (a² - ab + b²) = a³ - a^2b + ab² )

2. ( b \cdot (a² - ab + b²) = ba² - bab + b³ = ba² + ab² + b³ )

Combining like terms from both distributions yields:

( a³ + b³ ).

This confirms that the expression simplifies correctly back to the original sum of cubes, thereby validating the formula. Understanding this factoring method is important in polynomial algebra, as it allows for simplification and solving of equations involving cubic polynomials effectively.

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