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Deriving the (a + b)³ Formula
The formula (a + b)³ = a³ + 3a²b + 3ab² + b³ is a fundamental concept in algebra, allowing us to expand the cube of a binomial expression. Here's how we can derive this formula:
Understanding the Concept
The expression (a + b)³ represents multiplying the binomial (a + b) by itself three times:
(a + b)³ = (a + b) * (a + b) * (a + b)
Step-by-Step Derivation
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Expand the first two terms:
(a + b) * (a + b) = a² + ab + ba + b² = a² + 2ab + b²
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Multiply the result by (a + b):
(a² + 2ab + b²) * (a + b) = a³ + 2a²b + ab² + a²b + 2ab² + b³
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Combine like terms:
a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + 3a²b + 3ab² + b³
Visualizing the Formula
The formula can be visualized using the following:
- Imagine a cube with sides of length (a + b).
- The volume of this cube is (a + b)³.
- This volume can be broken down into eight smaller cubes and six rectangular prisms.
- The eight smaller cubes represent a³ and b³.
- The six rectangular prisms represent 3a²b and 3ab².
Application in Algebra
The (a + b)³ formula is widely used in:
- Simplifying algebraic expressions: Expanding cubes of binomials efficiently.
- Solving equations: Using the formula to rewrite equations in a simpler form.
- Factorization: Recognizing and factoring expressions of the form a³ + 3a²b + 3ab² + b³.
Conclusion
The derivation of the (a + b)³ formula is a clear demonstration of applying distributive properties and combining like terms in algebra. This formula is crucial for understanding and manipulating algebraic expressions, making it a fundamental concept in mathematics.