## Analyzing the Rational Expression (x^2 + 6x + 12) / (x - 3)

This article explores the rational expression (x^2 + 6x + 12) / (x - 3), covering its key features, simplification, and potential applications.

### Understanding the Expression

The expression (x^2 + 6x + 12) / (x - 3) represents a **rational function**, which is a function defined as a ratio of two polynomials.

**Numerator:**x^2 + 6x + 12 is a quadratic polynomial.**Denominator:**x - 3 is a linear polynomial.

### Simplifying the Expression

The expression cannot be simplified further by factoring. This means the numerator and denominator do not share any common factors.

### Analyzing the Expression

**1. Domain:**
The function is defined for all real numbers except where the denominator equals zero.

- x - 3 = 0
- x = 3

Therefore, the **domain** of the function is all real numbers except x = 3. This value is referred to as a **vertical asymptote**.

**2. Vertical Asymptote:**
As x approaches 3, the denominator approaches zero, making the function approach infinity. This indicates a **vertical asymptote** at x = 3.

**3. Horizontal Asymptote:**
To find the **horizontal asymptote**, we compare the degrees of the numerator and denominator:

- The degree of the numerator (2) is greater than the degree of the denominator (1).

This indicates that there is **no horizontal asymptote**. The function will approach infinity as x approaches positive or negative infinity.

**4. Intercepts:**

**x-intercept:**To find the x-intercept, set the numerator equal to zero and solve:- x^2 + 6x + 12 = 0
- This quadratic equation does not have real roots. Therefore, there are
**no x-intercepts**.

**y-intercept:**To find the y-intercept, set x = 0:- (0^2 + 6(0) + 12) / (0 - 3) = -4

Therefore, the **y-intercept** is (0, -4).

### Graphing the Function

The graph of the function will exhibit the following characteristics:

**Vertical asymptote**at x = 3**No horizontal asymptote****y-intercept**at (0, -4)**No x-intercepts**

The graph will approach infinity as x approaches 3 and infinity as x approaches positive or negative infinity.

### Applications

Rational functions like (x^2 + 6x + 12) / (x - 3) can be used to model various real-world phenomena:

**Physics:**Describing the motion of objects under certain conditions.**Economics:**Modeling supply and demand curves.**Engineering:**Analyzing the behavior of circuits and systems.

Understanding the characteristics and behavior of this type of function is crucial for applying them to solve practical problems in various fields.