%5E3-expanded.jpeg "(a-b)^3 Expanded")
Expanding (a - b)³
The expression (a - b)³ represents the cube of the binomial (a - b). Expanding this expression involves multiplying the binomial by itself three times.
The Formula
The expanded form of (a - b)³ is:
(a - b)³ = a³ - 3a²b + 3ab² - b³
This formula can be derived using the distributive property of multiplication and the binomial theorem.
Derivation
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Square the binomial: (a - b)² = (a - b)(a - b) = a² - 2ab + b²
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Multiply the result by (a - b): (a - b)³ = (a² - 2ab + b²)(a - b) = a²(a - b) - 2ab(a - b) + b²(a - b)
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Distribute and simplify: = a³ - a²b - 2a²b + 2ab² + ab² - b³ = a³ - 3a²b + 3ab² - b³
Example
Let's expand (2x - 3y)³ using the formula:
(2x - 3y)³ = (2x)³ - 3(2x)²(3y) + 3(2x)(3y)² - (3y)³
= 8x³ - 36x²y + 54xy² - 27y³
Conclusion
Expanding (a - b)³ using the formula simplifies the process of finding the expanded form. This formula is widely used in algebra, calculus, and other branches of mathematics. Remember that the formula can be applied to any binomials, not just those involving variables 'a' and 'b'.