%5E3-formula-class-9.jpeg "(x+y+z)^3 Formula Class 9")
Cube of the Sum of Three Variables: (x+y+z)^3 Formula for Class 9
Introduction
In algebra, we often come across expressions that involve the sum of variables raised to a power. One such expression is (x+y+z)^3, where x, y, and z are variables. In this article, we will explore the formula for expanding (x+y+z)^3 and its application in real-world problems.
The Formula
The formula for (x+y+z)^3 is:
(x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2(y+z) + 3y^2(x+z) + 3z^2(x+y) + 6xyz
This formula is known as the cube of the sum of three variables formula.
Derivation of the Formula
To derive the formula, we can start by expanding (x+y+z)^3 using the binomial theorem:
(x+y+z)^3 = (x+y+z)(x+y+z)(x+y+z)
Expanding the product, we get:
(x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3xy^2 + 3xz^2 + 3y^2x + 3y^2z + 3yz^2 + 6xyz
Simplifying the expression, we get the formula:
(x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2(y+z) + 3y^2(x+z) + 3z^2(x+y) + 6xyz
Applications of the Formula
The (x+y+z)^3 formula has several applications in mathematics and other fields, including:
- Algebra: The formula is used to expand and simplify algebraic expressions.
- Geometry: The formula is used to calculate the volume of a cube with side length x+y+z.
- Physics: The formula is used to model real-world problems, such as the motion of objects in three dimensions.
Examples
- Expand (2x+3y+z)^3 using the formula:
(2x+3y+z)^3 = 2^3x^3 + 3^3y^3 + z^3 + 3(2x)^2(3y+z) + 3(3y)^2(2x+z) + 3z^2(2x+3y) + 6(2x)(3y)(z)
- Find the volume of a cube with side length x+y+z:
Volume = (x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2(y+z) + 3y^2(x+z) + 3z^2(x+y) + 6xyz
Conclusion
In this article, we have discussed the formula for (x+y+z)^3 and its application in algebra, geometry, and physics. The formula is a powerful tool for expanding and simplifying algebraic expressions and has numerous applications in real-world problems.
