Visualizing y = (x³ - 3x² + 2x) / (x - 1)

Let's break down this complex rational function into its components.Here's our function. It's a ratio of two polynomials.The function has two main parts: a cubic polynomial in the numerator, and a linear term in the denominator.Let's examine each term in the numerator. We have a cubic term x cubed, a quadratic term negative three x squared, and a linear term two x.The numerator can be factored step by step. First, we factor out x.Then we can factor the quadratic expression x squared minus three x plus two into two linear factors.The denominator is the linear term x minus one. It's crucial to note that x cannot equal one, as this would make the denominator zero.Here's our function in its fully factored form. This will be crucial for understanding its behavior.Now that we understand the components, we can analyze the function's key features.To find the holes and asymptotes, let's first examine our function.We start with our rational function y equals x cubed minus three x squared plus two x, all divided by x minus one.First, let's factor the numerator. We can pull out x as a common factor.The quadratic term can be further factored into x minus one times x minus two.Notice that x minus one appears in both numerator and denominator. After cancellation, we get y equals x times x minus two, but this is only valid when x is not equal to one.This cancellation reveals a hole in our function at x equals one. To find the y-coordinate of the hole, we can evaluate the simplified function as x approaches one.As x approaches infinity or negative infinity, our function approaches the line y equals x. This diagonal line is our horizontal asymptote.The function approaches but never touches the point at x equals one, creating a removable discontinuity. The graph continues smoothly on either side of the hole.