Select All The Correct Answers.Consider This Product:$\[ \frac{x^2-4x-1}{8x^2+6x} \cdot \frac{x^2+8x}{x^2+11x+24} \\]Which Values Are Excluded For The Product?- $-3$- 0- 2- $-8$- 7

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Introduction

When dealing with rational expressions, it's essential to understand the concept of excluded values. These values are the ones that make the denominator of the expression equal to zero, resulting in an undefined or infinite value. In this article, we will explore the excluded values for a given product of rational expressions and determine the correct answers.

Understanding Excluded Values

Excluded values are the values that make the denominator of a rational expression equal to zero. This is because division by zero is undefined in mathematics. When a rational expression has a denominator that equals zero, the expression is said to be undefined or infinite.

The Given Product of Rational Expressions

The given product of rational expressions is:

x2−4x−18x2+6x⋅x2+8xx2+11x+24\frac{x^2-4x-1}{8x^2+6x} \cdot \frac{x^2+8x}{x^2+11x+24}

Finding the Excluded Values

To find the excluded values, we need to set the denominators of each rational expression equal to zero and solve for x.

Denominator 1: 8x^2 + 6x

Setting the denominator equal to zero, we get:

8x2+6x=08x^2 + 6x = 0

Solving for x, we get:

x(8x+6)=0x(8x + 6) = 0

x=0 or x=−34x = 0 \text{ or } x = -\frac{3}{4}

Denominator 2: x^2 + 11x + 24

Setting the denominator equal to zero, we get:

x2+11x+24=0x^2 + 11x + 24 = 0

Solving for x, we get:

(x+8)(x+3)=0(x + 8)(x + 3) = 0

x=−8 or x=−3x = -8 \text{ or } x = -3

Determining the Excluded Values

Based on the solutions obtained, the excluded values for the product of rational expressions are:

  • x=0x = 0
  • x=−8x = -8
  • x=−3x = -3

Conclusion

In conclusion, the excluded values for the given product of rational expressions are x=0x = 0, x=−8x = -8, and x=−3x = -3. These values make the denominator of the expression equal to zero, resulting in an undefined or infinite value.

Final Answer

The correct answers are:

  • −3-3
  • 00
  • −8-8

These values are excluded for the product of the given rational expressions.

Introduction

In our previous article, we explored the concept of excluded values in a product of rational expressions. We determined the excluded values for a given product of rational expressions and provided the correct answers. In this article, we will address some frequently asked questions related to excluded values in a product of rational expressions.

Q&A

Q1: What are excluded values in a product of rational expressions?

A1: Excluded values are the values that make the denominator of a rational expression equal to zero. This is because division by zero is undefined in mathematics.

Q2: How do I find the excluded values for a product of rational expressions?

A2: To find the excluded values, you need to set the denominators of each rational expression equal to zero and solve for x.

Q3: What if I have a rational expression with multiple denominators?

A3: If you have a rational expression with multiple denominators, you need to set each denominator equal to zero and solve for x. The values that make any of the denominators equal to zero are the excluded values.

Q4: Can I have a rational expression with no excluded values?

A4: Yes, it is possible to have a rational expression with no excluded values. This occurs when the denominator of the expression is never equal to zero for any value of x.

Q5: How do I determine the excluded values for a rational expression with a quadratic denominator?

A5: To determine the excluded values for a rational expression with a quadratic denominator, you need to factor the denominator and set each factor equal to zero. The values that make any of the factors equal to zero are the excluded values.

Q6: Can I have a rational expression with multiple excluded values?

A6: Yes, it is possible to have a rational expression with multiple excluded values. This occurs when the denominator of the expression has multiple factors that can be equal to zero for different values of x.

Q7: How do I simplify a rational expression with excluded values?

A7: To simplify a rational expression with excluded values, you need to cancel out any common factors between the numerator and denominator. However, you should not cancel out any factors that would result in an excluded value.

Q8: Can I have a rational expression with a denominator that has no real solutions?

A8: Yes, it is possible to have a rational expression with a denominator that has no real solutions. This occurs when the denominator is a quadratic expression with no real roots.

Conclusion

In conclusion, excluded values are an essential concept in algebra, and understanding how to find and work with them is crucial for simplifying and solving rational expressions. By following the steps outlined in this article, you can determine the excluded values for a product of rational expressions and simplify the expression accordingly.

Final Answer

The correct answers to the frequently asked questions are:

  • Q1: Excluded values are the values that make the denominator of a rational expression equal to zero.
  • Q2: To find the excluded values, you need to set the denominators of each rational expression equal to zero and solve for x.
  • Q3: If you have a rational expression with multiple denominators, you need to set each denominator equal to zero and solve for x.
  • Q4: Yes, it is possible to have a rational expression with no excluded values.
  • Q5: To determine the excluded values for a rational expression with a quadratic denominator, you need to factor the denominator and set each factor equal to zero.
  • Q6: Yes, it is possible to have a rational expression with multiple excluded values.
  • Q7: To simplify a rational expression with excluded values, you need to cancel out any common factors between the numerator and denominator.
  • Q8: Yes, it is possible to have a rational expression with a denominator that has no real solutions.