(x-2)(x-2).jpeg "(x-2)(x-2)(x-2)")
Understanding (x-2)(x-2)(x-2)
The expression (x-2)(x-2)(x-2) is a polynomial, specifically a cubic polynomial. It represents the product of three identical binomials: (x-2). This type of expression has several important properties and can be explored in different ways.
Simplifying the Expression
One way to understand the expression is to simplify it through multiplication.
- Start with the first two binomials: (x-2)(x-2) = x² - 2x - 2x + 4 = x² - 4x + 4
- Multiply the simplified result by the third binomial: (x² - 4x + 4)(x-2) = x³ - 4x² + 4x - 2x² + 8x - 8 = x³ - 6x² + 12x - 8
Therefore, the simplified form of (x-2)(x-2)(x-2) is x³ - 6x² + 12x - 8.
Understanding the Roots
The expression (x-2)(x-2)(x-2) is also a factored form of the cubic polynomial. This factorization allows us to quickly identify its roots.
Roots are the values of x that make the polynomial equal to zero. Since the expression is factored, we can set each factor equal to zero and solve for x:
- x - 2 = 0 => x = 2
This means the expression has a repeated root at x = 2. The root has a multiplicity of 3 because the factor (x-2) appears three times.
Graphing the Expression
The simplified form, x³ - 6x² + 12x - 8, allows us to graph the expression. The graph will have the following features:
- Shape: The graph of a cubic function typically has an "S" shape.
- Turning Points: The graph will have two turning points.
- Root: The graph will intersect the x-axis at x = 2.
- Y-intercept: The graph will intersect the y-axis at y = -8 (found by setting x = 0 in the simplified form).
Applications
Understanding cubic expressions like (x-2)(x-2)(x-2) is important in various fields, including:
- Algebra: Solving equations, factoring polynomials, and understanding polynomial behavior.
- Calculus: Finding derivatives and integrals.
- Physics: Modeling physical phenomena like projectile motion or wave patterns.
- Engineering: Designing structures, analyzing circuits, and solving optimization problems.
Overall, understanding the expression (x-2)(x-2)(x-2) provides a foundation for understanding more complex mathematical concepts and their applications in real-world scenarios.