%5E3-formula-expansion.jpeg "(a-b-c)^3 Formula Expansion")
The (a-b-c)^3 Formula Expansion
The cube of a binomial expression (a-b-c) is a fundamental concept in algebra, and its expansion is a crucial formula to master. In this article, we will delve into the expansion of (a-b-c)^3, exploring the formula, its derivation, and examples to illustrate its application.
The Formula
The expansion of (a-b-c)^3 is given by:
(a-b-c)^3 = a^3 - 3a^2b + 3ab^2 - b^3 - 3a^2c + 6abc - 3b^2c - c^3
This formula might look daunting at first, but it can be broken down and understood by analyzing the patterns of the terms.
Derivation
To derive the formula, we can start by using the cube of a binomial formula:
(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
Now, let's add and subtract c to both sides of the equation:
(a-b-c)^3 = ((a-b)-c)^3 = ((a-b)^3 - c^3 - 3(a-b)^2c + 3(a-b)bc - c^3)
Expanding the expression and simplifying the terms, we arrive at the final formula:
(a-b-c)^3 = a^3 - 3a^2b + 3ab^2 - b^3 - 3a^2c + 6abc - 3b^2c - c^3
Examples
Let's put the formula into practice with some examples:
Example 1
Expand (x-2y-3z)^3:
(x-2y-3z)^3 = x^3 - 3x^2(2y) + 3x(2y)^2 - (2y)^3 - 3x^2(3z) + 6x(2y)(3z) - 3(2y)^2(3z) - (3z)^3
Simplifying the expression, we get:
x^3 - 6x^2y + 12xy^2 - 8y^3 - 9x^2z + 36xyz - 27y^2z - 27z^3
Example 2
Expand (2a-b-4c)^3:
(2a-b-4c)^3 = (2a)^3 - 3(2a)^2b + 3(2a)b^2 - b^3 - 3(2a)^2(4c) + 6(2a)b(4c) - 3b^2(4c) - (4c)^3
Simplifying the expression, we get:
8a^3 - 12a^2b + 6ab^2 - b^3 - 48a^2c + 96abc - 24b^2c - 64c^3
Conclusion
The (a-b-c)^3 formula expansion is a powerful tool for algebraic manipulations. By understanding the formula and its derivation, you can expand complex expressions involving the cube of a binomial. Practice exercises and examples will help solidify your grasp of this fundamental concept.
