(x+1)^3-(x-1)(x^2+x+1)-2=0

2 min read Jun 16, 2024

Solving the Equation: (x+1)^3 - (x-1)(x^2+x+1) - 2 = 0

This article aims to solve the equation (x+1)^3 - (x-1)(x^2+x+1) - 2 = 0. We will use algebraic manipulation and the properties of exponents and multiplication to simplify the equation and find its solution(s).

Expanding the Equation

First, we expand the equation using the following rules:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
(a - b)(a^2 + ab + b^2) = a^3 - b^3

Applying these rules, we get:

(x+1)^3 - (x-1)(x^2+x+1) - 2 = 0 => x^3 + 3x^2 + 3x + 1 - (x^3 - 1) - 2 = 0

Simplifying the Equation

Now, we simplify the equation by combining like terms:

x^3 + 3x^2 + 3x + 1 - x^3 + 1 - 2 = 0 => 3x^2 + 3x = 0

Solving for x

To solve for x, we factor out a 3x from the equation:

3x(x + 1) = 0

This gives us two possible solutions:

3x = 0 => x = 0
x + 1 = 0 => x = -1

Conclusion

Therefore, the solutions to the equation (x+1)^3 - (x-1)(x^2+x+1) - 2 = 0 are x = 0 and x = -1.