%5E3%3Da%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3.jpeg "(a+b)^3=a^3+3a^2b+3ab^2+b^3")
Understanding the Expansion of (a + b)³
The expansion of the binomial (a + b)³ is a fundamental concept in algebra with numerous applications in various fields.
The Expansion:
The expansion of (a + b)³ is given by:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Derivation and Explanation:
This expansion can be derived by multiplying (a + b) by itself three times:
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(a + b)³ = (a + b) * (a + b) * (a + b)
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Expand the first two brackets: (a + b) * (a + b) = a² + 2ab + b²
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Multiply the result by (a + b): (a² + 2ab + b²) * (a + b) = a³ + 3a²b + 3ab² + b³
Visual Representation:
The expansion can be visualized using a binomial cube. Imagine a cube with each side having length (a + b). The volume of this cube is (a + b)³.
The volume can also be calculated by adding the volumes of all the smaller cubes and rectangular prisms that make up the larger cube. These smaller volumes correspond to the terms in the expansion:
- a³: represents the volume of a cube with side length 'a'.
- 3a²b: represents the volume of three rectangular prisms with dimensions 'a' x 'a' x 'b'.
- 3ab²: represents the volume of three rectangular prisms with dimensions 'a' x 'b' x 'b'.
- b³: represents the volume of a cube with side length 'b'.
Applications:
The expansion of (a + b)³ has numerous applications in various fields, including:
- Algebraic manipulation: It is used in simplifying complex expressions and solving equations.
- Calculus: It plays a crucial role in calculating derivatives and integrals.
- Physics: It is used in deriving formulas for work, energy, and momentum.
Conclusion:
The expansion of (a + b)³ is a fundamental algebraic formula with wide-ranging applications. Understanding its derivation and visual representation enhances its comprehension and allows for its effective application in various contexts.