%2F(x%5E2-4).jpeg "(18x^3-3x^2+x-1)/(x^2-4)")
Analyzing the Rational Expression: (18x^3 - 3x^2 + x - 1) / (x^2 - 4)
This article will explore the rational expression (18x^3 - 3x^2 + x - 1) / (x^2 - 4). We will delve into its properties, perform simplification, and discuss its domain and potential for further analysis.
Understanding the Expression
The expression represents a rational function, a function that can be expressed as the ratio of two polynomials.
- Numerator: The polynomial 18x^3 - 3x^2 + x - 1
- Denominator: The polynomial x^2 - 4
Simplifying the Expression
The expression can be simplified using polynomial long division or synthetic division.
1. Polynomial Long Division:
18x - 3
x^2-4 | 18x^3 - 3x^2 + x - 1
-(18x^3 - 72x)
------------------
-3x^2 + 73x - 1
-(-3x^2 + 12)
------------------
73x - 13
Therefore, (18x^3 - 3x^2 + x - 1) / (x^2 - 4) simplifies to 18x - 3 + (73x - 13) / (x^2 - 4).
2. Synthetic Division:
Since the denominator is a quadratic, synthetic division can be used as well. The process is similar to polynomial long division, but it uses a more compact notation.
Note: The simplified form can be further factored, but it doesn't significantly alter the analysis of the expression.
Domain of the Expression
The domain of a rational expression is all real numbers except for the values that make the denominator equal to zero.
- Finding the excluded values:
- x^2 - 4 = 0
- (x + 2)(x - 2) = 0
- x = -2, x = 2
Therefore, the domain of the expression is all real numbers except for x = -2 and x = 2.
Further Analysis
The simplified expression allows for a deeper understanding of the rational function:
- Asymptotes: The expression has a vertical asymptote at x = 2 and x = -2, due to the denominator becoming zero at these points. The expression also has an oblique asymptote, as the degree of the numerator is one greater than the degree of the denominator, which can be determined from the simplified form (18x - 3).
- Holes: There are no holes in the graph of this rational function.
- Intercepts: The x and y-intercepts can be calculated by setting the numerator and denominator equal to zero, respectively.
Conclusion
This article provided a comprehensive analysis of the rational expression (18x^3 - 3x^2 + x - 1) / (x^2 - 4). We explored simplification techniques, determined its domain, and highlighted key features for further analysis. Understanding these aspects allows for a deeper comprehension of the behavior of the function and its graph.