(x^2-1)^3 Expand

Jasonbradley

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Jun 17, 2024

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Expanding (x² - 1)³

Expanding expressions with exponents can be a bit tricky, but with the right methods, it becomes manageable. Let's explore how to expand (x² - 1)³.

Understanding the Basics

The expression (x² - 1)³ is simply (x² - 1) multiplied by itself three times:

(x² - 1)³ = (x² - 1)(x² - 1)(x² - 1)

Using the Binomial Theorem

One way to expand this is by using the Binomial Theorem. This theorem provides a formula for expanding expressions of the form (a + b)ⁿ:

(a + b)ⁿ = aⁿ + nCa¹aⁿ⁻¹b¹ + nC₂aⁿ⁻²b² + ... + nCn⁻¹abⁿ⁻¹ + bⁿ

Where nCk represents the binomial coefficient, which is calculated as:

nCk = n! / (k! * (n - k)!)

Applying this to our expression, we have:

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

Simplifying, we get:

(x² - 1)³ = x⁶ - 3x⁴ + 3x² - 1

Expanding by Multiplication

Alternatively, we can expand the expression by direct multiplication:

1. Start with the first two factors: (x² - 1)(x² - 1) = x⁴ - 2x² + 1

2. Multiply the result by the remaining factor: (x⁴ - 2x² + 1)(x² - 1) = x⁶ - 2x⁴ + x² - x⁴ + 2x² - 1

3. Combine like terms: x⁶ - 3x⁴ + 3x² - 1

The Result

Both methods lead to the same expanded form: (x² - 1)³ = x⁶ - 3x⁴ + 3x² - 1

This expression is now in its simplest form, ready for further manipulation or analysis.