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Exploring the Power of (x-3)(x-3)(x-3)
This expression, often written as (x-3)³, represents a fascinating concept in algebra and has significant applications in various fields. Let's delve into its meaning and explore its uses.
Unveiling the Cube
(x-3)³ signifies the product of (x-3) multiplied by itself three times:
(x-3)³ = (x-3) * (x-3) * (x-3)
This expression represents a cube in mathematics, where the base, (x-3), is multiplied by itself a certain number of times, in this case, three times.
Expanding the Expression
We can expand the expression using the distributive property or by using the binomial theorem:
(x-3)³ = (x-3) * (x-3) * (x-3)
Step 1: Expand the first two terms: (x-3) * (x-3) = x² - 6x + 9
Step 2: Multiply the result from Step 1 by (x-3): (x² - 6x + 9) * (x-3) = x³ - 9x² + 27x - 27
Therefore, (x-3)³ = x³ - 9x² + 27x - 27
Applications in Various Fields
Understanding (x-3)³ has relevance in various fields, including:
- Algebra: It is used in solving equations, simplifying expressions, and understanding polynomial functions.
- Calculus: The derivative of (x-3)³ is 3(x-3)² and the integral is (1/4)(x-3)⁴, which play vital roles in various calculus concepts.
- Geometry: The expression can be used to find the volume of a cube with side length (x-3).
- Physics: The expression can be applied in analyzing motion with constant acceleration.
Conclusion
(x-3)³ is a powerful expression that encapsulates the concept of cubing a binomial and has extensive applications in mathematics, calculus, geometry, and physics. It highlights the interconnectedness of different areas of knowledge and the beauty of mathematical concepts.