(x+1)^2(x-1)^2

Jasonbradley

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Jun 16, 2024

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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.