- A rational expression's excluded values occur where the denominator equals zero.
- Given the excluded values \(-3, 0,\) and \(8\), the denominator must contain the factors \((x+3), x,\) and \((x-8)\).
- Testing each denominator shows that only the first expression, \(\frac{x+2}{x^3-5x^2-24x}\), results in these three specific values when the denominator is set to zero.
- The final expression is \(\boxed{\frac{x+2}{x^3-5x^2-24x}}\).
Explanation
- Analyzing Excluded Values and Denominators
To find the correct rational expression that has \(-3, 0,\) and \(8\) as its excluded values, we recall that the excluded values of a rational expression are precisely the values for which the denominator equals zero. If a rational expression has excluded values \(r_1, r_2, \dots, r_n\), then its denominator \(Q(x)\) must be a polynomial that vanishes at each of these points. This means the denominator must contain the factors \((x - r_1), (x - r_2), \dots, (x - r_n)\). In our case, the excluded values are \(-3, 0,\) and \(8\), so the denominator must be a multiple of \((x - (-3))(x - 0)(x - 8) = (x+3) \cdot x \cdot (x-8) = x(x^2 - 5x - 24) = x^3 - 5x^2 - 24x\).
- Testing Each Expression
Now we examine the denominator of each provided option:
- \(\frac{x+2}{x^3-5x^2-24x}\): The denominator is \(x^3 - 5x^2 - 24x\). Factoring out \(x\), we get \(x(x^2 - 5x - 24)\). We then factor the quadratic expression \(x^2 - 5x - 24\) by finding two numbers that multiply to \(-24\) and add to \(-5\). These numbers are \(-8\) and \(3\). So, \(x^2 - 5x - 24 = (x-8)(x+3)\). Thus, the full denominator is \(x(x-8)(x+3)\). Setting this to zero gives the excluded values \(x=0, x=8,\) and \(x=-3\). This matches the given set.
- \(\frac{x+2}{x^2-5x-24}\): The denominator is \((x-8)(x+3)\). Setting this to zero gives the excluded values \(x=8\) and \(x=-3\). This does not include \(x=0\).
- \(\frac{x^3-5x^2-24x}{x+2}\): The denominator is \(x+2\). Setting this to zero gives the excluded value \(x=-2\). This does not match.
- \(\frac{x^2-5x-24}{x+2}\): The denominator is \(x+2\). Setting this to zero gives the excluded value \(x=-2\). This does not match.
- Conclusion and Final Answer
Comparing our results from Step 2, we see that the only expression with the set of excluded values \(\{-3, 0, 8\}\) is \(\frac{x+2}{x^3-5x^2-24x}\). Therefore, this is the correct rational expression. The final answer is \(\boxed{\frac{x+2}{x^3-5x^2-24x}}\).
Examples
Rational expressions are used in engineering and physics to model rates and physical systems where constraints exist. For example, if you are designing a circuit where the resistance \(R\) is given by a formula involving a variable input, you must identify the inputs (like frequency or voltage) that would cause the resistance to be undefined (the excluded values) to prevent system failure. Knowing how to find excluded values ensures that you operate within safe parameters.