(3x+2x^3-9-8x^2)/(x^2+1)

4 min read Jun 16, 2024

Analyzing the Expression: (3x + 2x^3 - 9 - 8x^2) / (x^2 + 1)

This expression represents a rational function, which is a function defined as the ratio of two polynomials. Let's break down the expression and analyze its properties:

1. Simplifying the Numerator:

The numerator is a polynomial with terms arranged in descending order of their exponents: 2x^3 - 8x^2 + 3x - 9. There are no common factors that can be factored out, so the numerator remains as is.

2. Analyzing the Denominator:

The denominator is a quadratic polynomial: x^2 + 1. This polynomial is notable because it is irreducible over real numbers. It cannot be factored into linear factors with real coefficients.

3. Understanding the Function:

The entire expression represents a function that takes an input value (x) and produces an output value based on the formula:

(2x^3 - 8x^2 + 3x - 9) / (x^2 + 1)

4. Domain of the Function:

The domain of a rational function is restricted by the values that make the denominator zero. However, in this case, x^2 + 1 is always positive for any real value of x. Therefore, the denominator is never zero, and the domain of this function is all real numbers.

5. Analyzing Asymptotes:

Vertical Asymptotes: Since the denominator is never zero, there are no vertical asymptotes.
Horizontal Asymptotes: The degree of the numerator (3) is higher than the degree of the denominator (2). This implies that there is no horizontal asymptote. Instead, the function will have a slant or oblique asymptote.
Slant Asymptote: To find the equation of the slant asymptote, we perform polynomial long division:

       2x - 8         
x^2+1 | 2x^3 - 8x^2 + 3x - 9 
        -(2x^3      + 2x)
        -----------------
             -8x^2 + x - 9
             -(-8x^2 - 8)
             -------------
                   x - 1

The quotient, 2x - 8, represents the equation of the slant asymptote.

6. Key Features:

The function has a slant asymptote at y = 2x - 8.
The function is defined for all real numbers.
The function may have local maximums and minimums, which can be found using calculus.

7. Further Exploration:

Graphing: By plotting the function, we can visually observe the behavior of the function, including its slant asymptote.
Calculus: Using calculus, we can determine the function's critical points, intervals of increase/decrease, and concavity.
Applications: Rational functions have various applications in fields like physics, engineering, and economics.

This analysis provides a foundation for further exploration of the function defined by the given expression.