3-formula-solution.jpeg "(a+b+c)3 Formula Solution")
The (a + b + c)³ Formula: A Comprehensive Guide
The formula for (a + b + c)³ is a fundamental concept in algebra that helps expand and simplify expressions. Understanding this formula is crucial for solving problems involving cubic equations, algebraic manipulations, and various mathematical applications.
Deriving the Formula
The formula is derived by expanding the cube of the trinomial:
(a + b + c)³ = (a + b + c)(a + b + c)(a + b + c)
Expanding this product involves multiplying each term in the first binomial by each term in the second, and then by each term in the third. This results in a total of 27 terms, but many of them are like terms. After combining like terms, we get the following formula:
(a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc
Understanding the Formula's Components
- a³, b³, c³: These terms represent the cubes of each individual variable.
- 3a²b, 3a²c, 3ab², 3ac², 3b²c, 3bc²: These terms represent the products of the squares of one variable multiplied by another variable, with a coefficient of 3.
- 6abc: This term represents the product of all three variables multiplied by 6.
Key Points to Remember
- Symmetry: The formula is symmetrical in a, b, and c. You can swap any two variables and the result will remain the same.
- Expansion: Expanding the cube using the formula saves time and prevents potential errors that could arise from multiplying the binomials directly.
- Applications: This formula is essential for solving equations involving cubic expressions, simplifying algebraic expressions, and analyzing various mathematical concepts.
Example Application
Let's say we need to expand the expression (2x + 3y - z)³. Using the formula, we get:
(2x + 3y - z)³ = (2x)³ + (3y)³ + (-z)³ + 3(2x)²(3y) + 3(2x)²(-z) + 3(2x)(3y)² + 3(2x)(-z)² + 3(3y)²(-z) + 3(3y)(-z)² + 6(2x)(3y)(-z)
Simplifying this expression gives:
(2x + 3y - z)³ = 8x³ + 27y³ - z³ + 36x²y - 12x²z + 54xy² - 6xz² - 27y²z - 9yz² - 36xyz
Conclusion
The (a + b + c)³ formula is a powerful tool for expanding and simplifying expressions involving cubes of trinomials. Understanding its derivation and application allows for efficient problem-solving in various mathematical contexts.