## Exploring the Polynomial (x+1)(x+2)(x-3)

This article will delve into the polynomial (x+1)(x+2)(x-3), analyzing its key features and exploring its behavior.

### Expanding the Expression

The first step in understanding this polynomial is to expand it. We can do this by using the distributive property (also known as FOIL) multiple times:

**(x+1)(x+2)(x-3) = **

Expanding the first two factors:

**(x+1)(x+2) = x² + 3x + 2**

Now, we multiply this result by (x-3):

**(x² + 3x + 2)(x-3) = x³ + 3x² + 2x - 3x² - 9x - 6**

Combining like terms, we get the expanded form of the polynomial:

**(x+1)(x+2)(x-3) = x³ - 7x - 6**

### Finding the Roots

The roots of a polynomial are the values of x that make the polynomial equal to zero. To find the roots of (x+1)(x+2)(x-3), we can simply set the expanded form equal to zero:

**x³ - 7x - 6 = 0**

We can then solve for x using various methods like factoring, the rational root theorem, or numerical methods. In this case, it's easy to see that the polynomial factors nicely:

**(x+1)(x+2)(x-3) = 0**

This gives us the following roots:

**x = -1****x = -2****x = 3**

These roots represent the x-intercepts of the polynomial's graph.

### Graphing the Polynomial

The graph of the polynomial (x+1)(x+2)(x-3) will be a curve that intersects the x-axis at the points (-1, 0), (-2, 0), and (3, 0).

Since the polynomial has a leading coefficient of 1 and an odd degree (3), the graph will have the following characteristics:

**As x approaches positive infinity, the graph will also approach positive infinity.****As x approaches negative infinity, the graph will approach negative infinity.****The graph will have a general "S" shape, with a change in direction at the x-intercepts.**

### Analyzing the Polynomial's Behavior

We can further analyze the polynomial's behavior by looking at its derivative. The derivative of the polynomial is:

**3x² - 7**

This derivative is equal to zero when x = ±√(7/3). These points represent the local extrema of the polynomial's graph.

By examining the sign of the derivative in different intervals, we can determine the intervals where the polynomial is increasing and decreasing:

**The polynomial is decreasing for x < -√(7/3) and x > √(7/3).****The polynomial is increasing for -√(7/3) < x < √(7/3).**

### Conclusion

The polynomial (x+1)(x+2)(x-3) is a cubic polynomial with roots at x = -1, x = -2, and x = 3. Its graph has a general "S" shape and exhibits both increasing and decreasing behavior. By understanding the key features of this polynomial, we can gain a deeper appreciation for its behavior and its relationship to other mathematical concepts.