(a+b)^3+(a-b)^3 Simplify

Jasonbradley

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Jun 16, 2024

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Simplifying (a+b)³ + (a-b)³

This expression can be simplified using the algebraic identities for the sum and difference of cubes:

Identities:

• Sum of Cubes: (a + b)³ = a³ + 3a²b + 3ab² + b³
• Difference of Cubes: (a - b)³ = a³ - 3a²b + 3ab² - b³

Simplifying the Expression:

1. Expand using the identities: (a + b)³ + (a - b)³ = (a³ + 3a²b + 3ab² + b³) + (a³ - 3a²b + 3ab² - b³)

2. Combine like terms: = a³ + 3a²b + 3ab² + b³ + a³ - 3a²b + 3ab² - b³ = 2a³ + 6ab²

3. Factor out common factors: = 2a(a² + 3b²)

Therefore, the simplified form of (a+b)³ + (a-b)³ is 2a(a² + 3b²).

Example:

Let's say a = 2 and b = 1.

• Original expression: (2 + 1)³ + (2 - 1)³ = 3³ + 1³ = 27 + 1 = 28
• Simplified expression: 2(2)(2² + 3(1²)) = 4(4 + 3) = 4(7) = 28

Conclusion:

We have successfully simplified the given expression by using the identities for the sum and difference of cubes, combining like terms, and factoring out common factors. This simplified form allows for easier computation and analysis of the expression.