(x+2)^3+(x-2)^3-2x(x^2+12)

Jasonbradley

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Jun 16, 2024

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Exploring the Expression: (x+2)^3 + (x-2)^3 - 2x(x^2 + 12)

This article will explore the algebraic expression (x+2)^3 + (x-2)^3 - 2x(x^2 + 12). We will simplify the expression and discuss its properties.

Simplifying the Expression

We can simplify the expression using the following steps:

1. Expand the cubes:

   • (x+2)^3 = (x+2)(x+2)(x+2) = x^3 + 6x^2 + 12x + 8
   • (x-2)^3 = (x-2)(x-2)(x-2) = x^3 - 6x^2 + 12x - 8

2. Substitute the expanded expressions: (x^3 + 6x^2 + 12x + 8) + (x^3 - 6x^2 + 12x - 8) - 2x(x^2 + 12)

3. Simplify: 2x^3 + 24x - 2x^3 - 24x = 0

Therefore, the simplified form of the expression is 0.

Conclusion

The expression (x+2)^3 + (x-2)^3 - 2x(x^2 + 12) simplifies to 0. This means that the expression is equal to zero regardless of the value of x. This is a constant expression, and its value will always be zero.

This analysis demonstrates how algebraic manipulations can be used to simplify complex expressions and reveal hidden relationships.