(x^3-8x+6)/(x^2-2x+1)

4 min read Jun 17, 2024

Analyzing the Rational Function: (x³ - 8x + 6) / (x² - 2x + 1)

This article will explore the rational function (x³ - 8x + 6) / (x² - 2x + 1), delving into its key characteristics like domain, vertical asymptotes, horizontal asymptotes, and intercepts. Understanding these elements provides a comprehensive view of the function's behavior and its graphical representation.

1. Domain

The domain of a rational function is restricted by the values that make the denominator zero. To find these values, we factor the denominator:

x² - 2x + 1 = (x - 1)²

The denominator is zero when x = 1. Therefore, the domain of the function is all real numbers except for x = 1. In set notation, this is:

Domain: {x ∈ ℝ | x ≠ 1}

2. Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator becomes zero, but the numerator does not. In this case, the denominator is zero at x = 1, and the numerator is not zero at this point. Therefore, there is a vertical asymptote at x = 1.

3. Horizontal Asymptotes

The degree of the numerator (3) is greater than the degree of the denominator (2). This indicates that there is no horizontal asymptote. Instead, the function will have oblique asymptotes.

4. Oblique Asymptotes

To find the oblique asymptote, we perform long division of the numerator by the denominator.

        x + 2
x² - 2x + 1 | x³ - 8x + 6
              -(x³ - 2x² + x)
              -----------------
                    2x² - 9x + 6
                    -(2x² - 4x + 2)
                    ---------------
                         -5x + 4

The quotient of the division is x + 2, which represents the oblique asymptote of the function.

5. Intercepts

a. x-intercepts:

To find the x-intercepts, we set the numerator equal to zero and solve for x:

x³ - 8x + 6 = 0

This equation does not have an easy solution by factoring. We can use numerical methods (like the Newton-Raphson method) or graphing calculators to find approximate values of the x-intercepts.

b. y-intercept:

To find the y-intercept, we set x = 0 and solve for y:

y = (0³ - 8(0) + 6) / (0² - 2(0) + 1) = 6

Therefore, the y-intercept is (0, 6).

Conclusion

By analyzing the domain, vertical asymptotes, horizontal asymptotes, oblique asymptotes, and intercepts, we have gained a significant understanding of the behavior of the rational function (x³ - 8x + 6) / (x² - 2x + 1). This information allows us to accurately sketch its graph and predict its behavior at different points. Remember that the function has no horizontal asymptotes but does have an oblique asymptote at y = x + 2, and it also has a vertical asymptote at x = 1.