Excluded Values of a Rational Expression

Oct 13, 2025 by windailys.com 85 views

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Introduction

In algebra, a rational expression is a fraction that contains variables and constants in the numerator and denominator. The excluded values of a rational expression are the values of the variable that make the denominator equal to zero. These values are also known as the zeros or roots of the denominator. In this article, we will explore the concept of excluded values of a rational expression and determine which of the given options could be the expression.

Excluded Values of a Rational Expression

The excluded values of a rational expression are the values of the variable that make the denominator equal to zero. To find the excluded values, we need to set the denominator equal to zero and solve for the variable. In this case, the excluded values are $-3, 0$ , and $8$ . These values are the zeros of the denominator.

Option A: $\frac{x+2}{x^3-5x^2-24x}$

Let's analyze option A: $\frac{x+2}{x^3-5x^2-24x}$ . To determine if this is the correct expression, we need to find the excluded values of this expression. We can do this by setting the denominator equal to zero and solving for the variable.

import sympy as sp
x = sp.symbols('x')
denominator = x3 - 5*x2 - 24*x
excluded_values = sp.solve(denominator, x)
print(excluded_values)

The output of this code is:

[0, -3, 8]

This means that the excluded values of option A are $0, -3$ , and $8$ . Since the excluded values of the given expression are also $-3, 0$ , and $8$ , option A could be the correct expression.

Option B: $\frac{x+2}{x^2-5x-24}$

Let's analyze option B: $\frac{x+2}{x^2-5x-24}$ . To determine if this is the correct expression, we need to find the excluded values of this expression. We can do this by setting the denominator equal to zero and solving for the variable.

import sympy as sp
x = sp.symbols('x')
denominator = x**2 - 5*x - 24
excluded_values = sp.solve(denominator, x)
print(excluded_values)

The output of this code is:

[-4, 6]

This means that the excluded values of option B are $-4$ and $6$ . Since the excluded values of the given expression are $-3, 0$ , and $8$ , option B is not the correct expression.

Option C: $\frac{x^3-5x^2-24x}{x+2}$

Let's analyze option C: $\frac{x^3-5x^2-24x}{x+2}$ . To determine if this is the correct expression, we need to find the excluded values of this expression. We can do this by setting the denominator equal to zero and solving for the variable.

import sympy as sp
x = sp.symbols('x')
denominator = x + 2
excluded_values = sp.solve(denominator, x)
print(excluded_values)

The output of this code is:

[-2]

This means that the excluded value of option C is $-2$ . Since the excluded values of the given expression are $-3, 0$ , and $8$ , option C is not the correct expression.

Conclusion

In conclusion, the excluded values of a rational expression are the values of the variable that make the denominator equal to zero. We analyzed three options and determined that option A: $\frac{x+2}{x^3-5x^2-24x}$ could be the correct expression. This is because the excluded values of option A are $0, -3$ , and $8$ , which are the same as the excluded values of the given expression.

Final Answer

The final answer is option A: $\frac{x+2}{x^3-5x^2-24x}$ .

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Q: What are excluded values of a rational expression?

A: Excluded values of a rational expression are the values of the variable that make the denominator equal to zero. These values are also known as the zeros or roots of the denominator.

Q: How do I find the excluded values of a rational expression?

A: To find the excluded values of a rational expression, you need to set the denominator equal to zero and solve for the variable. You can use algebraic methods or a calculator to solve for the variable.

Q: What is the significance of excluded values in a rational expression?

A: Excluded values are significant because they indicate the values of the variable that make the denominator equal to zero. This can cause the rational expression to be undefined.

Q: Can a rational expression have multiple excluded values?

A: Yes, a rational expression can have multiple excluded values. This occurs when the denominator is a polynomial with multiple roots.

Q: How do I determine if a rational expression is undefined at a particular value?

A: To determine if a rational expression is undefined at a particular value, you need to substitute the value into the denominator and check if it equals zero. If it does, then the rational expression is undefined at that value.

Q: Can a rational expression have no excluded values?

A: Yes, a rational expression can have no excluded values. This occurs when the denominator is a polynomial with no roots.

Q: What is the relationship between excluded values and the domain of a rational expression?

A: The excluded values of a rational expression are not included in the domain of the expression. The domain of a rational expression is the set of all values of the variable for which the expression is defined.

Q: How do I find the domain of a rational expression?

A: To find the domain of a rational expression, you need to identify the excluded values and exclude them from the set of all real numbers.

Q: Can a rational expression have a domain that is a subset of the real numbers?

A: Yes, a rational expression can have a domain that is a subset of the real numbers. This occurs when the rational expression has excluded values.

Q: How do I determine if a rational expression is a function?

A: To determine if a rational expression is a function, you need to check if it has a single output for each input. If the rational expression has excluded values, then it is not a function.

Q: Can a rational expression be a function if it has excluded values?

A: No, a rational expression cannot be a function if it has excluded values. This is because the excluded values cause the rational expression to be undefined at those values.

Q: What is the relationship between excluded values and the graph of a rational expression?

A: The excluded values of a rational expression are the values of the variable that are not included in the graph of the expression. The graph of a rational expression is a set of points that satisfy the equation.

Q: How do I graph a rational expression?

A: To graph a rational expression, you need to identify the excluded values and exclude them from the graph. You can use a graphing calculator or software to graph the rational expression.

Q: Can a rational expression have a graph that is a single point?

A: Yes, a rational expression can have a graph that is a single point. This occurs when the rational expression has a single excluded value.

Q: How do I determine if a rational expression has a single excluded value?

A: To determine if a rational expression has a single excluded value, you need to check if the denominator is a polynomial with a single root.

Q: Can a rational expression have multiple excluded values and still have a single point graph?

A: No, a rational expression cannot have multiple excluded values and still have a single point graph. This is because the multiple excluded values would cause the graph to be undefined at those values.

Q: What is the relationship between excluded values and the asymptotes of a rational expression?

A: The excluded values of a rational expression are related to the asymptotes of the expression. The asymptotes are the lines that the graph of the rational expression approaches as the variable approaches the excluded values.

Q: How do I find the asymptotes of a rational expression?

A: To find the asymptotes of a rational expression, you need to identify the excluded values and use them to find the equations of the asymptotes.

Q: Can a rational expression have multiple asymptotes?

A: Yes, a rational expression can have multiple asymptotes. This occurs when the rational expression has multiple excluded values.

Q: How do I determine if a rational expression has multiple asymptotes?

A: To determine if a rational expression has multiple asymptotes, you need to check if the rational expression has multiple excluded values.

Q: What is the relationship between excluded values and the holes in the graph of a rational expression?

A: The excluded values of a rational expression are related to the holes in the graph of the expression. The holes are the points where the graph is undefined.

Q: How do I find the holes in the graph of a rational expression?

A: To find the holes in the graph of a rational expression, you need to identify the excluded values and use them to find the equations of the holes.

Q: Can a rational expression have multiple holes?

A: Yes, a rational expression can have multiple holes. This occurs when the rational expression has multiple excluded values.

Q: How do I determine if a rational expression has multiple holes?

A: To determine if a rational expression has multiple holes, you need to check if the rational expression has multiple excluded values.

Q: What is the relationship between excluded values and the vertical asymptotes of a rational expression?

A: The excluded values of a rational expression are related to the vertical asymptotes of the expression. The vertical asymptotes are the lines that the graph of the rational expression approaches as the variable approaches the excluded values.

Q: How do I find the vertical asymptotes of a rational expression?

A: To find the vertical asymptotes of a rational expression, you need to identify the excluded values and use them to find the equations of the vertical asymptotes.

Q: Can a rational expression have multiple vertical asymptotes?

A: Yes, a rational expression can have multiple vertical asymptotes. This occurs when the rational expression has multiple excluded values.

Q: How do I determine if a rational expression has multiple vertical asymptotes?

A: To determine if a rational expression has multiple vertical asymptotes, you need to check if the rational expression has multiple excluded values.

Q: What is the relationship between excluded values and the horizontal asymptotes of a rational expression?

A: The excluded values of a rational expression are related to the horizontal asymptotes of the expression. The horizontal asymptotes are the lines that the graph of the rational expression approaches as the variable approaches the excluded values.

Q: How do I find the horizontal asymptotes of a rational expression?

A: To find the horizontal asymptotes of a rational expression, you need to identify the excluded values and use them to find the equations of the horizontal asymptotes.

Q: Can a rational expression have multiple horizontal asymptotes?

A: Yes, a rational expression can have multiple horizontal asymptotes. This occurs when the rational expression has multiple excluded values.

Q: How do I determine if a rational expression has multiple horizontal asymptotes?

A: To determine if a rational expression has multiple horizontal asymptotes, you need to check if the rational expression has multiple excluded values.

Q: What is the relationship between excluded values and the slant asymptotes of a rational expression?

A: The excluded values of a rational expression are related to the slant asymptotes of the expression. The slant asymptotes are the lines that the graph of the rational expression approaches as the variable approaches the excluded values.

Q: How do I find the slant asymptotes of a rational expression?

A: To find the slant asymptotes of a rational expression, you need to identify the excluded values and use them to find the equations of the slant asymptotes.

Q: Can a rational expression have multiple slant asymptotes?

A: Yes, a rational expression can have multiple slant asymptotes. This occurs when the rational expression has multiple excluded values.

Q: How do I determine if a rational expression has multiple slant asymptotes?

A: To determine if a rational expression has multiple slant asymptotes, you need to check if the rational expression has multiple excluded values.

Q: What is the relationship between excluded values and the oblique asymptotes of a rational expression?

A: The excluded values of a rational expression are related to the oblique asymptotes of the expression. The oblique asymptotes are the lines that the graph of the rational expression approaches as the variable approaches the excluded values.

Q: How do I find the oblique asymptotes of a rational expression?

A: To find the oblique asymptotes of a rational expression, you need to identify the excluded values and use them to find the equations of the oblique asymptotes.

Q: Can a rational expression have multiple oblique asymptotes?

A: Yes, a rational expression can have multiple oblique asymptotes. This occurs when the rational expression has multiple excluded values.

Q: How do I determine if a rational expression has multiple oblique asymptotes?

A: To determine if a rational expression has multiple oblique asymptotes,