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

Recognize the pattern: We have the same factor repeated three times, which indicates we can use the concept of exponents.

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:

Expand the first two factors: (x  1)(x  1) = x²  2x + 1

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 xaxis 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.