((x^(2)-5)(x^(2)-4))/((x-1))

4 min read Jun 16, 2024

Analyzing the Rational Expression: ((x^(2)-5)(x^(2)-4))/((x-1))

This expression represents a rational function, which is a function defined as the ratio of two polynomials. To understand it better, let's break it down:

1. Factoring the Numerator:

The numerator, (x^(2)-5)(x^(2)-4), is already factored into two quadratic expressions.
We can further factor these expressions:
- (x^(2)-5) cannot be factored further over the real numbers.
- (x^(2)-4) can be factored as (x+2)(x-2) using the difference of squares pattern.

Therefore, the fully factored numerator is (x^(2)-5)(x+2)(x-2).

2. Simplifying the Expression:

The expression now becomes: ((x^(2)-5)(x+2)(x-2))/((x-1))

Notice that there are no common factors between the numerator and denominator. This means the expression cannot be simplified further.

3. Identifying Key Features:

Domain: The function is defined for all real numbers except where the denominator is zero. This occurs when x = 1. Therefore, the domain is (-∞, 1) U (1, ∞).
Vertical Asymptotes: The function has a vertical asymptote at x = 1 since the denominator approaches zero as x approaches 1.
Horizontal Asymptotes: To determine the horizontal asymptote, we need to consider the degrees of the numerator and denominator:
- The degree of the numerator is 4 (highest power of x is x^4).
- The degree of the denominator is 1 (highest power of x is x^1).
- Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, the function will have oblique asymptote.
x-intercepts: The x-intercepts occur when the numerator is zero. This happens when x = ±√5, x = -2, and x = 2.
y-intercept: The y-intercept occurs when x = 0. This gives us y = 20.

4. Graphing the Function:

Using the information above, we can sketch a rough graph of the function. It will have the following characteristics:

Vertical asymptote at x = 1.
x-intercepts at x = ±√5, x = -2, and x = 2.
y-intercept at y = 20.
No horizontal asymptote, but an oblique asymptote.

Conclusion:

This rational expression represents a function with several interesting features. Understanding its factorization, domain, asymptotes, and intercepts helps us visualize its behavior and understand its properties. While it cannot be simplified further, its complex structure allows for a rich analysis and leads to a unique graph.