(x^2+x-17)/(x-4)

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Jun 12, 2024

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Rational Expression: (x^2+x-17)/(x-4)

In this article, we will explore the rational expression (x^2+x-17)/(x-4). We will simplify the expression, find its domain, and analyze its behavior.

Simplification

To simplify the expression, we can start by factoring the numerator:

(x^2+x-17) = (x+5)(x-3)

Now, we can rewrite the expression as:

((x+5)(x-3))/(x-4)

As we can see, the numerator is already factored, and we can cancel out any common factors between the numerator and the denominator.

Domain

The domain of the expression is all real numbers except where the denominator is zero. In this case, the denominator is zero when x = 4. Therefore, the domain of the expression is:

{x | x ≠ 4}

Behavior

The expression (x^2+x-17)/(x-4) has a vertical asymptote at x = 4. As x approaches 4 from the left or right, the expression approaches positive or negative infinity.

When x is close to 0, the expression approaches -17/4. When x is large (positive or negative), the expression approaches x.

Graph

The graph of the expression (x^2+x-17)/(x-4) is a hyperbola with a vertical asymptote at x = 4. The graph opens upwards and has a minimum point at x = -5.

Here's a rough sketch of the graph:

      +
     / \
    /   \
   /     \
  /       \
 - - - - - - 
  |       |
  |  Minimum  |
  |       |
  - - - - - -

In conclusion, we have simplified the rational expression (x^2+x-17)/(x-4), found its domain, and analyzed its behavior. We have also sketched the graph of the expression.