(a%5E2-ab%2Bb%5E2).jpeg "(a+b)(a^2-ab+b^2)")
Understanding the Expansion of (a + b)(a² - ab + b²)
The expression (a + b)(a² - ab + b²) is a special case of polynomial multiplication that results in a very important formula. Let's break it down step-by-step:
The Concept
This expression represents the multiplication of two binomials:
- (a + b): The first binomial, containing two terms: 'a' and 'b'.
- (a² - ab + b²): The second binomial, containing three terms: 'a²', '-ab', and 'b²'.
To expand this, we'll use the distributive property of multiplication.
The Expansion
We can expand the expression by multiplying each term in the first binomial with every term in the second binomial:
- a(a² - ab + b²) = a³ - a²b + ab²
- b(a² - ab + b²) = a²b - ab² + b³
Now, combining the results:
(a + b)(a² - ab + b²) = a³ - a²b + ab² + a²b - ab² + b³
Notice that the terms -a²b and a²b, and ab² and -ab² cancel each other out.
The Result
This leaves us with the simplified form:
(a + b)(a² - ab + b²) = a³ + b³
Significance of the Formula
This formula is known as the sum of cubes formula. It provides a simple way to factorize expressions in the form of a³ + b³, and it has applications in various areas of mathematics, including algebra, trigonometry, and calculus.
In Summary:
The expansion of (a + b)(a² - ab + b²) demonstrates the power of distributive property and leads to the important sum of cubes formula: a³ + b³. This formula provides a crucial tool for simplifying and factoring expressions, contributing to a deeper understanding of mathematical concepts.