## Analyzing the Rational Expression: (4x^4 + 5x - 4) / (x^2 - 3x - 2)

This article will delve into the analysis of the rational expression (4x^4 + 5x - 4) / (x^2 - 3x - 2). We will explore its key components, simplification techniques, and the implications of its structure.

### Understanding the Components

**Numerator:**4x^4 + 5x - 4 is a polynomial of degree 4.**Denominator:**x^2 - 3x - 2 is a polynomial of degree 2.

### Simplifying the Expression

The first step in analyzing this expression is attempting to simplify it. This can be done by factoring both the numerator and denominator:

**Factoring the Denominator**

The denominator can be factored into: (x - 4)(x + 1/2).

**Factoring the Numerator**

Unfortunately, the numerator does not have simple integer roots, making it challenging to factor directly. We can use polynomial long division or synthetic division to find a possible factor.

**Polynomial Long Division**

By performing long division with (x^2 - 3x - 2) as the divisor and (4x^4 + 5x - 4) as the dividend, we find that:

(4x^4 + 5x - 4) / (x^2 - 3x - 2) = **4x^2 + 12x + 44** + (178x + 172) / (x^2 - 3x - 2)

**Simplified Expression**

Therefore, the simplified form of the expression is: **4x^2 + 12x + 44 + (178x + 172) / (x^2 - 3x - 2)**

### Implications of the Structure

**Degree of the Expression:**The degree of the numerator (4) is greater than the degree of the denominator (2). This means the expression is**top-heavy**and can be further simplified by long division or synthetic division.**Potential Discontinuities:**The original expression is undefined when the denominator is zero. Setting the denominator to zero and solving for x, we find that x = 4 and x = -1/2. These are**vertical asymptotes**of the expression, representing points where the function becomes unbounded.**Asymptotic Behavior:**The simplified expression highlights that the function will exhibit parabolic behavior (4x^2 + 12x + 44) with the addition of a rational term that approaches zero as x approaches infinity. This implies that there is a**horizontal asymptote**at y = 0.

### Conclusion

The expression (4x^4 + 5x - 4) / (x^2 - 3x - 2) represents a rational function with interesting properties. We have explored its simplified form, identified potential discontinuities, and analyzed its asymptotic behavior. This understanding is crucial for understanding the function's behavior, particularly its values and how it interacts with the x and y axes.