Select All The Correct Answers.Consider This Product:${ \frac{x 2-4x-21}{3x 2+6x} \cdot \frac{x 2+8x}{x 2+11x+24} }$Which Values Are Excluded Values For The Product?A. 2 B. 7 C. { -3$}$ D. 0 E. { -8$}$

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Understanding Excluded Values

When working with rational expressions, it's essential to identify the values that make the expression undefined. These values are known as excluded values or restrictions. In the context of rational expressions, excluded values occur when the denominator of the expression is equal to zero.

The Product of Rational Expressions

The given product of rational expressions is:

x2−4x−213x2+6x⋅x2+8xx2+11x+24\frac{x^2-4x-21}{3x^2+6x} \cdot \frac{x^2+8x}{x^2+11x+24}

To find the excluded values for this product, we need to consider the values that make each denominator equal to zero.

Finding Excluded Values

Let's start by finding the excluded values for the first denominator, 3x2+6x3x^2+6x. We can factor out the greatest common factor (GCF), which is 3x3x:

3x2+6x=3x(x+2)3x^2+6x = 3x(x+2)

Now, we can see that the first denominator will be equal to zero when x=0x=0 or x=−2x=-2.

Next, let's consider the second denominator, x2+11x+24x^2+11x+24. We can factor this quadratic expression as:

x2+11x+24=(x+8)(x+3)x^2+11x+24 = (x+8)(x+3)

Now, we can see that the second denominator will be equal to zero when x=−8x=-8 or x=−3x=-3.

Combining Excluded Values

To find the excluded values for the product, we need to combine the excluded values from each denominator. In this case, the excluded values are:

  • x=0x=0 (from the first denominator)
  • x=−2x=-2 (from the first denominator)
  • x=−8x=-8 (from the second denominator)
  • x=−3x=-3 (from the second denominator)

Conclusion

In conclusion, the excluded values for the product of rational expressions are x=0x=0, x=−2x=-2, x=−8x=-8, and x=−3x=-3. These values make the expression undefined, and we must exclude them when working with the product.

Answer Key

Based on the analysis above, the correct answers are:

  • A. 2 is not an excluded value, so it is incorrect.
  • B. 7 is not an excluded value, so it is incorrect.
  • C. −3-3 is an excluded value, so it is correct.
  • D. 0 is an excluded value, so it is correct.
  • E. −8-8 is an excluded value, so it is correct.

The final answer is: C,D,E\boxed{C, D, E}

Understanding Excluded Values

When working with rational expressions, it's essential to identify the values that make the expression undefined. These values are known as excluded values or restrictions. In the context of rational expressions, excluded values occur when the denominator of the expression is equal to zero.

Q&A: Excluded Values for the Product of Rational Expressions

Q: What are excluded values in the context of rational expressions?

A: Excluded values are the values that make the denominator of a rational expression equal to zero. These values make the expression undefined.

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

A: To find excluded values, you need to set the denominator equal to zero and solve for the variable. This will give you the values that make the denominator equal to zero.

Q: What is the difference between excluded values and zeros of a rational expression?

A: The zeros of a rational expression are the values that make the numerator equal to zero. Excluded values, on the other hand, are the values that make the denominator equal to zero.

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 product of multiple factors, and each factor can be equal to zero.

Q: How do I determine if a value is an excluded value for a rational expression?

A: To determine if a value is an excluded value, you need to substitute the value into the denominator and check if it equals zero. If it does, then the value is an excluded value.

Q: Can I simplify a rational expression by canceling out common factors in the numerator and denominator?

A: Yes, you can simplify a rational expression by canceling out common factors in the numerator and denominator. However, be careful not to cancel out any factors that are also excluded values.

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

A: To find the excluded values for a product of rational expressions, you need to find the excluded values for each individual rational expression and then combine them.

Q: Can I use the same method to find excluded values for a quotient of rational expressions?

A: Yes, you can use the same method to find excluded values for a quotient of rational expressions. However, you need to be careful to consider the signs of the numerators and denominators.

Conclusion

In conclusion, excluded values are an essential concept in working with rational expressions. By understanding how to find excluded values and how to use them to simplify rational expressions, you can become more confident and proficient in your math skills.

Answer Key

Based on the Q&A above, the correct answers are:

  • Q1: Excluded values are the values that make the denominator of a rational expression equal to zero.
  • Q2: To find excluded values, you need to set the denominator equal to zero and solve for the variable.
  • Q3: The zeros of a rational expression are the values that make the numerator equal to zero, while excluded values are the values that make the denominator equal to zero.
  • Q4: Yes, a rational expression can have multiple excluded values.
  • Q5: To determine if a value is an excluded value, you need to substitute the value into the denominator and check if it equals zero.
  • Q6: Yes, you can simplify a rational expression by canceling out common factors in the numerator and denominator.
  • Q7: To find the excluded values for a product of rational expressions, you need to find the excluded values for each individual rational expression and then combine them.
  • Q8: Yes, you can use the same method to find excluded values for a quotient of rational expressions.

The final answer is: Yes\boxed{Yes}