(x%5E2%2Bxy%2By%5E2)-formula.jpeg "(x-y)(x^2+xy+y^2) Formula")
Expanding the Formula: (x-y)(x^2+xy+y^2)
In algebra, expanding formulas is an essential skill to master. One of the most common and useful formulas to expand is (x-y)(x^2+xy+y^2). In this article, we will delve into the step-by-step process of expanding this formula and explore its applications.
The Formula: (x-y)(x^2+xy+y^2)
The given formula is a product of two binomials: (x-y) and (x^2+xy+y^2). Our goal is to expand this product using the distributive property of multiplication over addition.
Step-by-Step Expansion
Step 1: Multiply the First Term
Begin by multiplying the first term of the first binomial (x-y) with the entire second binomial (x^2+xy+y^2).
(x)(x^2+xy+y^2) = x^3 + x^2y + xy^2
Step 2: Multiply the Second Term
Next, multiply the second term of the first binomial (x-y) with the entire second binomial (x^2+xy+y^2).
(-y)(x^2+xy+y^2) = -x^2y - xy^2 - y^3
Step 3: Combine the Results
Now, combine the results of Steps 1 and 2.
(x-y)(x^2+xy+y^2) = x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3
Simplified Formula
After combining the terms, we can see that some terms cancel out. The resulting simplified formula is:
(x-y)(x^2+xy+y^2) = x^3 - y^3
Applications and Examples
The expanded formula (x-y)(x^2+xy+y^2) = x^3 - y^3 has numerous applications in various mathematical fields, such as:
- Factoring Cubic Expressions: This formula can be used to factorize cubic expressions of the form
x^3 - y^3. - Algebraic Manipulations: It can be used to simplify complex algebraic expressions and equations.
- Geometry and Trigonometry: This formula has applications in solving problems involving right triangles and trigonometric identities.
In conclusion, expanding the formula (x-y)(x^2+xy+y^2) requires a systematic approach using the distributive property of multiplication over addition. The resulting simplified formula x^3 - y^3 has numerous applications in various mathematical fields, making it an essential tool for any math enthusiast.
