(x-1)^3-(x+1)^3+6(x+1)(x-1)

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression (x-1)³ - (x+1)³ + 6(x+1)(x-1)

This article will guide you through simplifying the expression (x-1)³ - (x+1)³ + 6(x+1)(x-1). We will utilize algebraic identities and factorization to reach a simplified form.

Understanding the Problem

The given expression involves:

• Cubic terms: (x-1)³ and (x+1)³ represent cubes of binomial expressions.
• Product of binomials: 6(x+1)(x-1) is the product of two binomials.

Simplifying using Algebraic Identities

We can utilize the following algebraic identities to simplify the expression:

• Difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)
• Product of sum and difference: (a + b)(a - b) = a² - b²

Step 1: Applying the difference of cubes identity

Let's apply the difference of cubes identity to the first two terms:

(x-1)³ - (x+1)³ = [(x-1) - (x+1)][(x-1)² + (x-1)(x+1) + (x+1)²]

Step 2: Simplifying the resulting expression

Simplifying the expression within the square brackets:

[(x-1) - (x+1)][(x-1)² + (x-1)(x+1) + (x+1)²] = (-2)[(x²-2x+1) + (x²-1) + (x²+2x+1)]

Step 3: Applying the product of sum and difference identity

Now, let's apply the product of sum and difference identity to the third term:

6(x+1)(x-1) = 6(x² - 1)

Step 4: Combining the simplified terms

Combining the simplified terms from steps 2 and 3:

(-2)[(x²-2x+1) + (x²-1) + (x²+2x+1)] + 6(x² - 1)

Step 5: Expanding and simplifying further

Expanding and simplifying the expression:

-2(3x² + 1) + 6x² - 6 = -6x² - 2 + 6x² - 6 = -8

Conclusion

By applying algebraic identities and simplifying the expression, we have successfully simplified (x-1)³ - (x+1)³ + 6(x+1)(x-1) to -8. This demonstrates the power of utilizing algebraic identities for efficient simplification of complex expressions.