%5E3-formula.jpeg "(x-y-z)^3 Formula")
The Formula for (x-y-z)^3: A Comprehensive Guide
The formula for (x-y-z)^3 is a fundamental concept in algebra and is widely used in various mathematical operations. In this article, we will delve into the formula, its derivation, and some examples to illustrate its application.
The Formula
The formula for (x-y-z)^3 is given by:
(x-y-z)^3 = x^3 - 3x^2(y+z) + 3x(y^2 + z^2 + yz) - (y-z)^3
This formula is also known as the cube of a difference of three terms or the trinomial cube formula.
Derivation of the Formula
To derive the formula, we can start with the cube of a binomial, which is given by:
(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
Now, let's consider the expression (x-y-z)^3. We can rewrite it as:
(x - (y+z))^3
Using the binomial cube formula, we can expand the expression as:
(x - (y+z))^3 = x^3 - 3x^2(y+z) + 3x(y+z)^2 - (y+z)^3
Expanding the expression further, we get:
(x-y-z)^3 = x^3 - 3x^2(y+z) + 3x(y^2 + z^2 + 2yz) - (y^3 + z^3 - 3yz(y+z))
Simplifying the expression, we arrive at the final formula:
(x-y-z)^3 = x^3 - 3x^2(y+z) + 3x(y^2 + z^2 + yz) - (y-z)^3
Examples
Example 1:
Expand (2-3-4)^3 using the formula.
Using the formula, we get:
(2-3-4)^3 = 2^3 - 3(2)^2(3+4) + 3(2)(3^2 + 4^2 + 3*4) - (3-4)^3
Simplifying the expression, we get:
(2-3-4)^3 = -125
Example 2:
Simplify (x-2y+z)^3 using the formula.
Using the formula, we get:
(x-2y+z)^3 = x^3 - 3x^2(2y-z) + 3x((2y)^2 + z^2 + 2y*z) - (2y-z)^3
Simplifying the expression, we get:
(x-2y+z)^3 = x^3 - 6x^2y + 3x^2z + 12xy^2 - 6xyz + 3xz^2 - 8y^3 + 12y^2z - 6yz^2 + z^3
These examples illustrate the application of the formula for (x-y-z)^3 in simplifying complex expressions.
Conclusion
In conclusion, the formula for (x-y-z)^3 is a powerful tool in algebra, which can be used to simplify complex expressions involving the cube of a difference of three terms. The formula has numerous applications in various branches of mathematics, including algebra, geometry, and calculus. By mastering this formula, students and professionals alike can solve a wide range of mathematical problems with ease and accuracy.
