(a%5E2%2Bab%2Bb%5E2).jpeg "(a-b)(a^2+ab+b^2)")
Understanding the Expansion of (a - b)(a² + ab + b²)
The expression (a - b)(a² + ab + b²) is a classic example of a difference of cubes factorization. This particular pattern appears frequently in algebra and can be expanded using a simple rule.
The Difference of Cubes Formula
The general formula for the difference of cubes is:
a³ - b³ = (a - b)(a² + ab + b²)
This formula shows that the difference of two cubes can be factored into a binomial (a - b) and a trinomial (a² + ab + b²).
Expanding the Expression
To expand (a - b)(a² + ab + b²), we simply apply the distributive property:
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Multiply the first term of the binomial (a) by each term in the trinomial:
- a * a² = a³
- a * ab = a²b
- a * b² = ab²
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Multiply the second term of the binomial (-b) by each term in the trinomial:
- -b * a² = -a²b
- -b * ab = -ab²
- -b * b² = -b³
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Combine the resulting terms:
- a³ + a²b + ab² - a²b - ab² - b³ = a³ - b³
Conclusion
Therefore, expanding (a - b)(a² + ab + b²) using the difference of cubes formula results in a³ - b³. This pattern is crucial for simplifying expressions and solving equations involving the difference of two cubes. Understanding this factorization will prove invaluable for various mathematical operations and problem-solving situations.