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(x + 2y)³: A Comprehensive Guide
What is (x + 2y)³?
(x + 2y)³ is an algebraic expression that represents the cube of the sum of two variables, x and 2y. In other words, it is the result of multiplying the sum of x and 2y by itself three times.
Expanding (x + 2y)³
To expand (x + 2y)³, we can use the binomial theorem, which states that:
(a + b)³ = a³ + 3a²b + 3ab² + b³
In this case, a = x and b = 2y. Substituting these values into the formula, we get:
(x + 2y)³ = x³ + 3x²(2y) + 3x(2y)² + (2y)³
Simplifying the expression, we get:
(x + 2y)³ = x³ + 6x²y + 12xy² + 8y³
Properties of (x + 2y)³
(x + 2y)³ has several important properties that make it a useful expression in various mathematical contexts:
- Symmetry: The expression is symmetric about the point (0, 0), meaning that if we swap x and 2y, the expression remains unchanged.
- Cubic nature: The expression has a cubic nature, meaning that it grows rapidly as x and 2y increase.
- Factorization: The expression can be factorized into the product of three binomials: (x + 2y)³ = (x + 2y)(x² - 2xy + 4y²)(x - 2y).
Real-World Applications of (x + 2y)³
(x + 2y)³ has numerous applications in various fields, including:
- Physics: The expression is used to model the motion of objects in three-dimensional space.
- Engineering: It is used to design and optimize systems, such as electronic circuits and mechanical systems.
- Computer Science: The expression is used in computer graphics, game development, and scientific computing.
Conclusion
(x + 2y)³ is a fundamental expression in algebra that has far-reaching applications in various fields. Understanding its properties, expansion, and real-world applications can help us better appreciate the beauty and importance of mathematics in our daily lives.
