(x+1)^2(x-2)^2

4 min read Jun 16, 2024

Exploring the Function (x+1)²(x-2)²

This article explores the function (x+1)²(x-2)², uncovering its key characteristics and analyzing its behavior.

Understanding the Basics

The function (x+1)²(x-2)² is a polynomial function with a degree of 4. This means it has a maximum of four real roots (where the function crosses the x-axis). We can identify these roots directly from the factored form:

x = -1 (with a multiplicity of 2)
x = 2 (with a multiplicity of 2)

The multiplicity of the roots indicates how many times the graph touches the x-axis at that point. Since both roots have a multiplicity of 2, the graph will touch but not cross the x-axis at these points.

Analyzing the Function's Behavior

Symmetry: The function is symmetric about the line x= 1/2. This is because the exponents of both factors are even.
End Behavior: As x approaches positive or negative infinity, the function will approach positive infinity. This is because the leading term of the expanded form of the function is x⁴, which has a positive coefficient.
Local Extrema: The function has two local minima, both located at the roots x = -1 and x = 2. This is due to the even multiplicities of the roots.

Graphing the Function

To visualize the behavior of the function, we can plot its graph. It will have the following characteristics:

The graph will touch the x-axis at x = -1 and x = 2, forming "bounces" at these points.
The graph will approach positive infinity as x approaches positive or negative infinity.
There will be two minima, one at x = -1 and another at x = 2.

The graph of (x+1)²(x-2)² resembles a "W" shape, with the two minima forming the valleys.

Key Takeaways

(x+1)²(x-2)² is a polynomial function with a degree of 4.
It has two double roots at x = -1 and x = 2.
The function is symmetric about the line x = 1/2.
The function has two local minima, one at each root.

This analysis provides a comprehensive understanding of the function (x+1)²(x-2)², its key characteristics, and its behavior.