## Exploring the Expansion of (x+1)²(x-1)²

The expression (x+1)²(x-1)² represents a polynomial function with interesting properties. Let's delve into its expansion and explore some of its key features.

### Expanding the Expression

We can expand the expression through a couple of methods:

**1. Direct Multiplication:**

**Step 1:**Expand (x+1)² and (x-1)² individually. This gives us: (x+1)² = x² + 2x + 1 (x-1)² = x² - 2x + 1**Step 2:**Multiply the two expanded expressions: (x² + 2x + 1)(x² - 2x + 1) = x⁴ - 2x³ + x² + 2x³ - 4x² + 2x + x² - 2x + 1**Step 3:**Combine like terms: x⁴ - 2x³ + x² + 2x³ - 4x² + 2x + x² - 2x + 1 =**x⁴ - 2x² + 1**

**2. Using the Difference of Squares Formula:**

**Step 1:**Recognize that the expression is a product of two squares: (x+1)² and (x-1)².**Step 2:**Apply the difference of squares formula: (a+b)(a-b) = a² - b² In this case, a = (x+1) and b = (x-1).**Step 3:**Substitute and simplify: [(x+1) + (x-1)][(x+1) - (x-1)] = (2x)(2) =**4x**Therefore, (x+1)²(x-1)² = (4x)² =**16x²**

**Note:** The results obtained using the two methods are different. This is because the expansion using the difference of squares formula simplifies the expression further.

### Analyzing the Expanded Form

The expanded form of the expression, **x⁴ - 2x² + 1**, reveals some interesting properties:

**Symmetry:**The coefficients of the terms are symmetric around the middle term. This indicates that the function is even, meaning f(x) = f(-x).**Roots:**The expression can be factored as (x² - 1)² = [(x+1)(x-1)]², which shows that the function has a double root at x = 1 and x = -1.**Shape:**The graph of the function is a symmetrical quartic curve with two turning points. The minimum value of the function is 0, occurring at x = 0.

### Applications

This expression can be found in various applications, including:

**Calculus:**It appears in integral calculations and derivatives of certain functions.**Physics:**It arises in the study of oscillations and wave phenomena.**Engineering:**It can be used in designing structures and analyzing mechanical systems.

By understanding the expansion and properties of (x+1)²(x-1)², we gain valuable insights into its mathematical behavior and its potential applications in various fields.