3-formula.jpeg "(x-3)3 Formula")
Understanding the (x - 3)³ Formula
The formula (x - 3)³ is a common algebraic expression used to represent the cube of the binomial (x - 3). Expanding this formula can be done using the following steps:
Expanding the Formula
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Understanding the Cube: The exponent '3' indicates that the binomial (x - 3) is multiplied by itself three times: (x - 3) * (x - 3) * (x - 3)
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Using the Distributive Property: We can expand this expression using the distributive property. It is crucial to multiply each term within the first binomial by each term within the second and third binomials.
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Simplifying the Expression: After multiplying all terms, combine similar terms to obtain a simplified form.
The Expanded Form
Following the above steps, we arrive at the expanded form of (x - 3)³:
(x - 3)³ = x³ - 9x² + 27x - 27
Understanding the Terms
- x³: This term results from multiplying x from each of the three binomials.
- -9x²: This term is obtained by combining the terms that involve multiplying x² and -3.
- 27x: This term arises from combining the terms that involve multiplying x and (-3)².
- -27: This is the constant term obtained by multiplying (-3) from each of the three binomials.
Applications of the Formula
The (x - 3)³ formula has applications in various areas of mathematics, including:
- Algebraic Manipulation: Simplifying expressions and solving equations.
- Calculus: Finding derivatives and integrals of functions.
- Geometry: Calculating volumes and areas.
- Physics: Modeling physical phenomena.
Example
Problem: Expand (2 - 3)³
Solution:
- Substitute x = 2 into the expanded formula: (2 - 3)³ = 2³ - 9(2)² + 27(2) - 27
- Simplify the expression: (2 - 3)³ = 8 - 36 + 54 - 27
- Calculate the final result: (2 - 3)³ = -1
Therefore, (2 - 3)³ = -1.
Conclusion
The (x - 3)³ formula is a fundamental algebraic expression with various applications. Understanding how to expand and apply this formula is crucial for success in mathematics and related fields.