(x^2+8x+1)/(x-4)

5 min read Jun 17, 2024

Analyzing the Rational Expression (x^2 + 8x + 1) / (x - 4)

This article will explore the rational expression (x^2 + 8x + 1) / (x - 4). We will examine its key features, including:

1. Domain

The domain of a rational expression is restricted by the values that make the denominator zero. In this case, the denominator (x - 4) becomes zero when x = 4. Therefore, the domain of the expression is all real numbers except x = 4.

2. Vertical Asymptote

A vertical asymptote occurs where the denominator of a rational expression approaches zero, and the numerator doesn't. Here, as x approaches 4, the denominator goes to zero, while the numerator approaches 41. This indicates a vertical asymptote at x = 4.

3. Horizontal Asymptote

To determine the horizontal asymptote, we consider the degrees of the numerator and denominator polynomials.

Case 1: Degree of numerator < Degree of denominator: The horizontal asymptote is at y = 0.
Case 2: Degree of numerator = Degree of denominator: The horizontal asymptote is at y = (leading coefficient of numerator) / (leading coefficient of denominator).
Case 3: Degree of numerator > Degree of denominator: There is no horizontal asymptote, but there might be a slant asymptote.

In our expression, the degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, there is no horizontal asymptote.

4. Slant Asymptote

Since the degree of the numerator is one higher than the degree of the denominator, we have a slant asymptote. To find it, we perform polynomial long division:

        x + 12
x - 4 | x^2 + 8x + 1 
          -(x^2 - 4x)
          ----------------
                12x + 1
                -(12x - 48)
                ----------------
                      49

The result of the division is x + 12 with a remainder of 49. The quotient, x + 12, represents the slant asymptote of the expression.

5. Intercepts

x-intercept: To find the x-intercept, set the expression equal to zero and solve for x. (x^2 + 8x + 1) / (x - 4) = 0 x^2 + 8x + 1 = 0 This quadratic equation does not factor easily. We can use the quadratic formula to find the x-intercepts.
y-intercept: To find the y-intercept, set x = 0 and evaluate the expression. (0^2 + 8(0) + 1) / (0 - 4) = -1/4

6. Graph

By combining the information about the domain, asymptotes, and intercepts, we can sketch a graph of the expression. The graph will have a vertical asymptote at x = 4, a slant asymptote at y = x + 12, and will pass through the point (0, -1/4).

Remember that the graph will not touch the vertical asymptote, and it will approach the slant asymptote as x approaches positive or negative infinity.

7. Conclusion

The rational expression (x^2 + 8x + 1) / (x - 4) exhibits several key features. It has a vertical asymptote at x = 4, a slant asymptote at y = x + 12, and a y-intercept at (0, -1/4). Understanding these features allows for a comprehensive analysis of the expression's behavior and its graphical representation.