(x-1)(x-1)(x-1)

Jasonbradley

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Jun 17, 2024

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Exploring the Expression (x-1)(x-1)(x-1)

The expression (x-1)(x-1)(x-1) is a simple yet interesting algebraic expression that can be analyzed and simplified. Let's delve into its properties and explore its meaning.

Understanding the Structure

This expression represents the product of three identical factors: (x-1). It is essentially the same as (x-1) cubed, or (x-1)³.

Expanding the Expression

To fully understand the expression, we can expand it using the distributive property:

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

2. Multiply the result by the third factor: (x² - 2x + 1)(x-1) = x³ - 2x² + x - x² + 2x - 1

3. Combine like terms: x³ - 3x² + 3x - 1

Therefore, the expanded form of (x-1)(x-1)(x-1) is x³ - 3x² + 3x - 1.

Recognizing the Pattern

Notice that the expanded form of (x-1)³ follows a specific pattern:

• The first term is x raised to the power of the exponent (3 in this case).
• The second term is the product of the exponent (3) and x raised to the power of one less than the exponent (2). The coefficient is negative.
• The third term is the product of the exponent (3) and x raised to the power of two less than the exponent (1). The coefficient is positive.
• The last term is 1 raised to the power of the exponent (3).

This pattern is a characteristic of expanding any binomial raised to a power.

Applications and Significance

Understanding the expansion of (x-1)³ has various applications:

• Factoring polynomials: By recognizing the pattern, we can factor polynomials of this form.
• Solving equations: If we set the expression equal to zero, we can solve for the roots of the equation.
• Graphing functions: The expression can be used to define a function whose graph can be analyzed and interpreted.

In summary, the expression (x-1)(x-1)(x-1) or (x-1)³ is a simple yet powerful algebraic expression that demonstrates the concept of multiplying binomials and reveals interesting patterns in the expansion. Its understanding is valuable in various mathematical contexts.