4 min read Jun 17, 2024

Analyzing the Expression: (x^3 - 4x + 6) / (x + 3)

This expression represents a rational function, a function defined as the ratio of two polynomials. Let's break down its components and explore its behavior.

The Numerator: x^3 - 4x + 6

  • Degree: The highest power of x is 3, making this a cubic polynomial.
  • Constant Term: The constant term is 6.
  • Roots: Finding the roots (where the polynomial equals zero) for cubic polynomials can be challenging. In this case, we can't easily factor it or apply the rational root theorem.

The Denominator: x + 3

  • Degree: The highest power of x is 1, making this a linear polynomial.
  • Zero: The denominator equals zero when x = -3.

Analyzing the Function's Behavior

  • Vertical Asymptote: Since the denominator becomes zero at x = -3, there is a vertical asymptote at this point. This means the function approaches infinity (or negative infinity) as x approaches -3 from either side.
  • Horizontal Asymptote: Because the degree of the numerator (3) is greater than the degree of the denominator (1), the function does not have a horizontal asymptote. Instead, it will have a slant (or oblique) asymptote.
  • Slant Asymptote: To find the slant asymptote, we need to perform polynomial long division or synthetic division. This process will yield a linear equation that represents the slant asymptote.
  • End Behavior: As x approaches positive or negative infinity, the function's behavior is dominated by the highest power terms. In this case, it will behave like x^3 / x = x^2. This means the function will grow increasingly large (or small) in a parabolic manner as x approaches infinity.

Graphing the Function

To get a complete understanding of the function's behavior, we can graph it. This graph will show the vertical asymptote, slant asymptote, and the general shape of the curve. Keep in mind that it's also useful to identify any x and y intercepts, which might require numerical methods or graphing calculators.

Additional Considerations

  • Domain: The function is defined for all real values except x = -3.
  • Range: Determining the exact range can be challenging, but we know it will include all real numbers except possibly some values near the vertical asymptote.

By understanding the components, behavior, and potential graph of this rational function, we can gain deeper insights into its characteristics and its behavior across different input values.

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