(a%5E2-ab%2Bb%5E2)-formula.jpeg "(a+b)(a^2-ab+b^2) Formula")
The (a + b)(a² - ab + b²) Formula: A Powerful Tool for Expansion
In algebra, the formula (a + b)(a² - ab + b²) is a fundamental concept that allows us to efficiently expand expressions and simplify complex equations. It's a special case of the difference of cubes factorization, and it's incredibly useful in various mathematical contexts.
Understanding the Formula
This formula states that the product of the sum of two terms (a + b) and the expression (a² - ab + b²) is equal to the cube of the first term (a³) plus the cube of the second term (b³).
Mathematically:
(a + b)(a² - ab + b²) = a³ + b³
How to Derive the Formula
We can derive this formula by applying the distributive property of multiplication:
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Expand the expression: (a + b)(a² - ab + b²) = a(a² - ab + b²) + b(a² - ab + b²)
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Distribute: = a³ - a²b + ab² + ba² - ab² + b³
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Simplify by combining like terms: = a³ + b³
The -a²b and +ba² terms cancel out, leaving us with the final result.
Applications of the Formula
The (a + b)(a² - ab + b²) formula finds applications in various mathematical areas, including:
- Factoring expressions: This formula can be used to factor expressions of the form a³ + b³.
- Solving equations: By applying this formula, we can simplify equations involving cubes and solve for unknown variables.
- Simplifying complex expressions: The formula can be used to simplify complex expressions that involve the product of (a + b) and (a² - ab + b²).
Example
Let's say we need to factor the expression x³ + 8. We can recognize that 8 is the cube of 2 (2³ = 8). Applying the formula, we get:
x³ + 8 = (x + 2)(x² - 2x + 4)
Conclusion
The (a + b)(a² - ab + b²) formula is a powerful tool in algebra that simplifies calculations and provides a shortcut for expanding and factoring expressions involving cubes. Its applications extend beyond basic algebra and prove useful in various mathematical disciplines.