%5E3-formula-class-9.jpeg "(x-y)^3 Formula Class 9")
Cube of a Difference Formula: (x-y)^3
In algebra, the cube of a difference formula is a mathematical expression that represents the cube of the difference between two variables. The formula is widely used in various mathematical operations, including simplification of expressions, factorization, and equation solving. In this article, we will discuss the cube of a difference formula, its expansion, and examples of its application.
What is the cube of a difference formula?
The cube of a difference formula is a mathematical expression that represents the cube of the difference between two variables x and y. It is denoted by (x-y)^3 and is defined as:
(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
This formula is used to expand the expression (x-y)^3 into its constituent terms.
Derivation of the cube of a difference formula
The cube of a difference formula can be derived by using the binomial theorem. The binomial theorem states that:
(a-b)^n = a^n - na^(n-1)b + (n(n-1))/2 a^(n-2)b^2 - ... + (-1)^n b^n
By substituting a = x and b = y, we get:
(x-y)^n = x^n - nx^(n-1)y + (n(n-1))/2 x^(n-2)y^2 - ... + (-1)^n y^n
Now, put n = 3 in the above equation to get:
(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
Examples of the cube of a difference formula
Example 1
Expand the expression (2x-3y)^3 using the cube of a difference formula.
(2x-3y)^3 = (2x)^3 - 3(2x)^2(3y) + 3(2x)(3y)^2 - (3y)^3 = 8x^3 - 36x^2y + 54xy^2 - 27y^3
Example 2
Simplify the expression (x-2)^3 + (x+2)^3 using the cube of a difference formula.
(x-2)^3 = x^3 - 3x^2(2) + 3x(2)^2 - 2^3 = x^3 - 6x^2 + 12x - 8 (x+2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3 = x^3 + 6x^2 + 12x + 8 (x-2)^3 + (x+2)^3 = 2x^3 + 24x = 2(x^3 + 12x)
Importance of the cube of a difference formula
The cube of a difference formula is an important concept in algebra and is used in various mathematical operations, including:
- Simplification of expressions: The formula is used to simplify complex expressions involving the cube of a difference.
- Factorization: The formula is used to factorize algebraic expressions involving the cube of a difference.
- Equation solving: The formula is used to solve equations involving the cube of a difference.
In conclusion, the cube of a difference formula is an essential concept in algebra that is used to expand the expression (x-y)^3 into its constituent terms. It has numerous applications in simplification of expressions, factorization, and equation solving.
