(x---1)(x---1).jpeg "(x - 1)(x - 1)(x - 1)")
Understanding the Expression (x - 1)(x - 1)(x - 1)
The expression (x - 1)(x - 1)(x - 1) is a polynomial expression that represents the product of three identical factors: (x - 1). Let's explore how to simplify and interpret this expression:
Simplifying the Expression
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Recognize the pattern: We have the same factor repeated three times, which indicates we can use the concept of exponents.
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Apply the exponent rule: Any factor multiplied by itself n times can be written as that factor raised to the power of n. In this case, (x - 1) multiplied by itself three times is equivalent to (x - 1)³.
Expanding the Expression
To fully understand the expression, we can expand it using the distributive property:
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Expand the first two factors: (x - 1)(x - 1) = x² - 2x + 1
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Multiply the result by the remaining factor: (x² - 2x + 1)(x - 1) = x³ - 3x² + 3x - 1
Interpretation
The expanded form, x³ - 3x² + 3x - 1, represents a cubic polynomial with:
- Degree: 3 (highest power of x)
- Leading coefficient: 1 (coefficient of the term with the highest power)
- Constant term: -1 (term without any x)
The expression (x - 1)³ indicates that this polynomial has a root at x = 1 with a multiplicity of 3. This means the graph of the function represented by this polynomial will touch the x-axis at x = 1 and then "bounce" back up or down, rather than crossing it.
Applications
Understanding this type of expression is important in various mathematical fields:
- Algebra: Solving equations, factoring, and simplifying expressions.
- Calculus: Finding derivatives and integrals.
- Graphing: Identifying key features of functions like roots and turning points.
In summary, (x - 1)³ is a simplified form of the expression (x - 1)(x - 1)(x - 1) representing a cubic polynomial with a root at x = 1 with a multiplicity of 3. Understanding this type of expression is essential for various mathematical applications.