## Exploring the Cubic Function: (x+1)(x-3)(x-4)

This article will delve into the properties and characteristics of the cubic function represented by the expression **(x+1)(x-3)(x-4)**. Understanding this function provides insights into its behavior, its graph, and its roots.

### Factorized Form and Roots

The given expression is already in its **factored form**. This makes it easy to identify the **roots** of the function. A root is a value of x that makes the function equal to zero.

**x+1 = 0**implies**x = -1****x-3 = 0**implies**x = 3****x-4 = 0**implies**x = 4**

Therefore, the function has **three roots**: -1, 3, and 4.

### Expanding the Expression

To understand the function's behavior further, we can expand the expression:

```
(x+1)(x-3)(x-4) = (x^2 - 2x - 3)(x-4)
= x^3 - 6x^2 + 5x + 12
```

The expanded form **x³ - 6x² + 5x + 12** provides information about the function's **degree**, **leading coefficient**, and **constant term**.

**Degree:**The highest power of x is 3, indicating a cubic function.**Leading Coefficient:**The coefficient of the x³ term is 1, which is positive. This means the function will have a**positive leading behavior**.**Constant Term:**The constant term is 12, which represents the y-intercept of the function's graph.

### Graphing the Function

The graph of this cubic function will exhibit the following characteristics:

**Roots:**The graph will intersect the x-axis at the points (-1, 0), (3, 0), and (4, 0).**Y-Intercept:**The graph will intersect the y-axis at the point (0, 12).**Leading Behavior:**Since the leading coefficient is positive, the graph will rise towards positive infinity as x approaches positive infinity and fall towards negative infinity as x approaches negative infinity.

The graph will have a **local maximum** between the roots -1 and 3, and a **local minimum** between the roots 3 and 4.

### Conclusion

The cubic function represented by **(x+1)(x-3)(x-4)** has three roots, a positive leading coefficient, and a constant term of 12. Its graph will exhibit a specific shape with local maxima and minima, and will intersect the x-axis at the roots and the y-axis at the constant term. By analyzing its factored form, expanded form, and graph, we gain valuable insights into the behavior of this cubic function.