3-formula-proof.jpeg "(a-b)3 Formula Proof")
Understanding the (a-b)³ Formula: A Comprehensive Proof
The formula (a-b)³ is a fundamental concept in algebra, particularly useful for simplifying expressions and solving equations. This article will provide a detailed proof of the formula and illustrate its application with examples.
The Formula:
The expansion of (a-b)³ is given by:
(a-b)³ = a³ - 3a²b + 3ab² - b³
Proof using the distributive property:
- Expand the cube: We can write (a-b)³ as (a-b)(a-b)(a-b).
- Apply distributive property: First, distribute (a-b) over the second (a-b):
- (a-b)*(a-b) = a(a-b) - b(a-b) = a² - ab - ba + b² = a² - 2ab + b²
- Distribute again: Now, distribute the result (a² - 2ab + b²) over the remaining (a-b):
- (a² - 2ab + b²)(a-b) = a(a² - 2ab + b²) - b(a² - 2ab + b²)
- Simplify: Expand the brackets and combine like terms:
- a³ - 2a²b + ab² - ba² + 2ab² - b³ = a³ - 3a²b + 3ab² - b³
Proof using binomial theorem:
The binomial theorem provides a general formula for expanding expressions of the form (x+y)ⁿ. We can apply it to (a-b)³ as follows:
- Binomial Theorem: (x+y)ⁿ = ∑ (n choose k) x^(n-k) y^k, where k ranges from 0 to n and (n choose k) is the binomial coefficient.
- Applying to (a-b)³: Substitute x=a, y=-b, and n=3 in the binomial theorem:
- (a-b)³ = (3 choose 0) a³ (-b)⁰ + (3 choose 1) a² (-b)¹ + (3 choose 2) a¹ (-b)² + (3 choose 3) a⁰ (-b)³
- Simplify: Calculate the binomial coefficients and simplify the terms:
- (a-b)³ = a³ - 3a²b + 3ab² - b³
Application of the formula:
The formula (a-b)³ has various applications, including:
- Simplifying expressions: It can be used to quickly simplify expressions involving cubes of differences.
- Solving equations: It can be used to factorize equations and find solutions.
- Calculus: It is used in deriving derivatives of expressions involving differences.
Example: Simplify the expression (2x - 3)³ using the formula.
Solution:
- Substitute a = 2x and b = 3 into the formula: (2x - 3)³ = (2x)³ - 3(2x)²(3) + 3(2x)(3)² - 3³
- Simplify the expression: = 8x³ - 36x² + 54x - 27
Therefore, the simplified expression is 8x³ - 36x² + 54x - 27.
In conclusion, the formula (a-b)³ is a powerful tool in algebra with numerous applications. Understanding its proof and applications can enhance problem-solving skills and simplify algebraic expressions.