(x-2)(x-3).jpeg "(x-1)(x-2)(x-3)")
Exploring the Polynomial (x-1)(x-2)(x-3)
The polynomial (x-1)(x-2)(x-3) is a fascinating expression that showcases several key concepts in algebra. Let's delve into its properties and explore its significance.
Expanding the Polynomial
First, let's expand the expression to understand its form:
(x-1)(x-2)(x-3) = (x^2 - 3x + 2)(x-3) = x^3 - 6x^2 + 11x - 6
We can observe that the expanded form is a cubic polynomial, meaning it has a highest power of x equal to 3.
Finding the Roots
The roots of a polynomial are the values of x for which the polynomial equals zero. In this case, we can easily find the roots by setting each factor equal to zero:
- x - 1 = 0 => x = 1
- x - 2 = 0 => x = 2
- x - 3 = 0 => x = 3
This highlights an important property: the roots of the polynomial are precisely the values that make each factor zero. This is a direct consequence of the Zero Product Property.
Graphing the Function
The polynomial (x-1)(x-2)(x-3) represents a cubic function. Its graph will have the following key features:
- Intercepts: The function crosses the x-axis at the points x = 1, x = 2, and x = 3. These are the roots we found earlier.
- End Behavior: As x approaches positive or negative infinity, the function will also approach positive or negative infinity respectively. This is because the leading term (x^3) dominates the expression.
- Shape: The graph will have two turning points, indicating a change in concavity.
Applications
The polynomial (x-1)(x-2)(x-3) can be used to model various real-world phenomena. For example, it can represent:
- The path of a projectile: The function can model the height of an object thrown vertically, with the roots representing the times when the object hits the ground.
- The volume of a container: The polynomial can represent the volume of a container with dimensions related to the variable x.
Conclusion
The polynomial (x-1)(x-2)(x-3) provides a simple yet powerful example of algebraic expressions. Its expansion, roots, and graphical representation illustrate fundamental concepts in algebra, showcasing the connections between different mathematical ideas. Understanding this polynomial lays the foundation for tackling more complex expressions and applying them to real-world problems.