Let's begin by simplifying the given function and then find the common factors, intercepts, and asymptotes.

Given function:

$r(x) = \frac{x^3 - x^2 - 2x}{x - 2}$

Step 1: Factor the numerator

We have: $x^3 - x^2 - 2x$ Factor out the common factor of $x$: $x(x^2 - x - 2)$ Now, factor $x^2 - x - 2$ further: $x^2 - x - 2 = (x - 2)(x + 1)$ Thus, the numerator becomes: $x(x - 2)(x + 1)$ Now the function is: $r(x) = \frac{x(x - 2)(x + 1)}{x - 2}$

Step 2: Cancel out the common factor

We notice that both the numerator and the denominator have the factor $x - 2$, so we can cancel it out: $r(x) = x(x + 1)$ However, we must remember that $x = 2$ is a point of discontinuity because it was cancelled, so there is a hole at $x = 2$.

Step 3: Find the intercepts

x-intercepts:

The x-intercepts occur when $r(x) = 0$. Set the numerator equal to zero: $x(x + 1) = 0$ Solve for $x$: $x = 0 \quad \text{or} \quad x = -1$ Thus, the x-intercepts are at $x = 0$ and $x = -1$.

y-intercept:

To find the y-intercept, set $x = 0$: $r(0) = 0(0 + 1) = 0$ Thus, the y-intercept is at $(0, 0)$.

Step 4: Asymptotes

Vertical Asymptote:

A vertical asymptote occurs when the denominator equals zero, and the factor is not cancelled. The denominator is $x - 2$, which would be zero at $x = 2$. Since this factor was cancelled, there is no vertical asymptote at $x = 2$, but there is a hole at $x = 2$.

Horizontal Asymptote:

As $x \to \infty$ or $x \to -\infty$, the degree of the numerator and denominator are both the same (degree 2 after simplification). The behavior of the function is dominated by the highest-degree term in both the numerator and denominator: $r(x) = x(x + 1) \approx x^2 \quad \text{as} \quad x \to \infty \text{ or } x \to -\infty$ Since the degree of the numerator is higher than the denominator, there is no horizontal asymptote. Instead, the function increases without bound.

Summary of Results:

• Common factors: $x - 2$ is the common factor that gets cancelled.
• Intercepts:
  • x-intercepts: $x = 0$ and $x = -1$
  • y-intercept: $(0, 0)$
• Asymptotes:
  • Vertical asymptote: None (but a hole at $x = 2$).
  • Horizontal asymptote: None.

Let me know if you would like more details or have further questions!

Related Questions:

1. How do you factor cubic polynomials like $x^3 - x^2 - 2x$?
2. What is the significance of the hole in the graph of a function?
3. Can a function have a vertical asymptote and a hole at the same point?
4. How do you find the horizontal asymptote of rational functions?
5. How does simplifying a rational expression affect its graph?

Tip: Always remember that after canceling a factor, check if that factor causes any discontinuity (like a hole), even if it's no longer in the simplified function.