## 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.