Introduction
The algebraic expression (a-b)³ (read as "a minus b whole cubed") is a fundamental cubic expansion formula. This guide breaks down its calculation, applications, and related mathematical concepts for students and professionals.
Step-by-Step Calculation
Step 1: Expand Using Binomial Theorem
The formula expands as:
[ (a-b)³ = (a-b)(a-b)(a-b) ]
Step 2: Simplify Stepwise
- First Multiplication:
[ (a-b)² = a² - 2ab + b² ] - Second Multiplication:
[ (a² - 2ab + b²)(a-b) = a³ - 3a²b + 3ab² - b³ ]
Final Result:
[ \boxed{(a-b)³ = a³ - 3a²b + 3ab² - b³} ]
Key Properties
- Symmetry: Mirrors the pattern of ((a+b)³) but alternates signs.
- Coefficients: Follow Pascal’s Triangle (1, 3, 3, 1).
- Applications: Used in polynomial factorization, calculus, and physics.
Common Mistakes to Avoid
- Sign Errors: Ensure negative terms are handled correctly.
- Misapplying Exponents: Remember ((a-b)³ \neq a³ - b³).
Related Formulas
| Formula | Expression |
|-----------------------|--------------------------------|
| Cube of Sum | ((a+b)³ = a³ + 3a²b + 3ab² + b³) |
| Difference of Cubes| (a³ - b³ = (a-b)(a² + ab + b²)) |
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FAQs
Q1: Why is ((a-b)³) not equal to (a³ - b³)?
A: The expansion involves middle terms ((-3a²b + 3ab²)), making it distinct from the simple subtraction of cubes.
Q2: How is this formula used in real-life scenarios?
A: It’s essential in engineering for stress calculations, economics for growth models, and computer algorithms.
Q3: Can this formula be applied to numbers with exponents?
A: Yes! Replace (a) and (b) with any algebraic terms (e.g., (x², y³)).
Conclusion
Mastering ((a-b)³) enhances problem-solving efficiency in algebra. Practice with real-world examples to solidify understanding.
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