## Exploring the Polynomial (x+1)(x-2)^2

The expression (x+1)(x-2)^2 represents a polynomial function. To understand its behavior, we can analyze its key features:

**1. Expanded Form:**

Firstly, let's expand the expression to get a clearer picture:

(x+1)(x-2)^2 = (x+1)(x^2 - 4x + 4)

Expanding further, we get:

**x^3 - 3x^2 - 4x + 4**

**2. Degree and Leading Coefficient:**

The degree of the polynomial is **3**, as the highest power of x is 3. This indicates it's a **cubic function**. The leading coefficient is **1**, which means the graph will rise to the right.

**3. Roots (x-intercepts):**

The roots of the polynomial are the values of x where the function equals zero. We can find them by setting the expanded form to zero:

x^3 - 3x^2 - 4x + 4 = 0

This equation can be factored as:

**(x+1)(x-2)^2 = 0**

This gives us the roots:

**x = -1**(multiplicity 1)**x = 2**(multiplicity 2)

The multiplicity of a root indicates how many times it appears in the factored form. A root with multiplicity 1 crosses the x-axis, while a root with multiplicity 2 touches the x-axis but doesn't cross it.

**4. End Behavior:**

As the leading coefficient is positive and the degree is odd, the end behavior of the function is as follows:

- As x approaches negative infinity, the function approaches negative infinity.
- As x approaches positive infinity, the function approaches positive infinity.

**5. Graphing:**

Using the information above, we can sketch a rough graph of the function:

- The graph will cross the x-axis at x = -1.
- The graph will touch the x-axis at x = 2.
- The graph will rise to the right and fall to the left.

**6. Applications:**

Polynomials like (x+1)(x-2)^2 have applications in various fields, including:

**Modeling real-world phenomena:**They can represent physical quantities like position, velocity, or acceleration over time.**Engineering:**They are used in designing structures, circuits, and systems.**Economics:**They can model economic growth, demand curves, and other economic relationships.

Understanding the properties and behavior of polynomials is crucial for solving problems in diverse areas. Analyzing the expanded form, roots, degree, and end behavior provides valuable insights into the function's characteristics.