For Which Rational Expressions Is -5 An Excluded Value? Select Two Options.A. { \frac{x+5}{x-5}$}$B. { \frac{x 2-5}{x 2+5}$}$C. { \frac{x-3}{x^2-25}$}$D. { \frac{2x+1}{x^2+25}$}$E. { \frac{x-5}{x^2+5x}$}$

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Understanding Rational Expressions and Excluded Values

Rational expressions are a fundamental concept in algebra, and understanding when they are undefined is crucial for solving equations and inequalities. In this article, we will explore the concept of excluded values in rational expressions and determine for which rational expressions -5 is an excluded value.

What are Excluded Values in Rational Expressions?

Excluded values in rational expressions are the values of the variable that make the denominator of the expression equal to zero. When the denominator is equal to zero, the expression is undefined, and the value is said to be an excluded value.

Why are Excluded Values Important?

Excluded values are important because they can affect the validity of the solution to an equation or inequality. If a solution is an excluded value, it is not a valid solution to the equation or inequality.

Determining Excluded Values

To determine the excluded values of a rational expression, we need to set the denominator equal to zero and solve for the variable. The values that make the denominator equal to zero are the excluded values.

Option A: {\frac{x+5}{x-5}$}$

For option A, we need to set the denominator equal to zero and solve for x.

x−5=0x - 5 = 0

x=5x = 5

Since x = 5 makes the denominator equal to zero, -5 is an excluded value for option A.

Option B: {\frac{x2-5}{x2+5}$}$

For option B, we need to set the denominator equal to zero and solve for x.

x2+5=0x^2 + 5 = 0

x2=−5x^2 = -5

Since x^2 cannot be equal to -5, there are no real solutions for x. Therefore, -5 is not an excluded value for option B.

Option C: {\frac{x-3}{x^2-25}$}$

For option C, we need to set the denominator equal to zero and solve for x.

x2−25=0x^2 - 25 = 0

(x+5)(x−5)=0(x + 5)(x - 5) = 0

x+5=0orx−5=0x + 5 = 0 \quad \text{or} \quad x - 5 = 0

x=−5orx=5x = -5 \quad \text{or} \quad x = 5

Since x = -5 makes the denominator equal to zero, -5 is an excluded value for option C.

Option D: {\frac{2x+1}{x^2+25}$}$

For option D, we need to set the denominator equal to zero and solve for x.

x2+25=0x^2 + 25 = 0

x2=−25x^2 = -25

Since x^2 cannot be equal to -25, there are no real solutions for x. Therefore, -5 is not an excluded value for option D.

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

For option E, we need to set the denominator equal to zero and solve for x.

x2+5x=0x^2 + 5x = 0

x(x+5)=0x(x + 5) = 0

x=0orx+5=0x = 0 \quad \text{or} \quad x + 5 = 0

x=0orx=−5x = 0 \quad \text{or} \quad x = -5

Since x = -5 makes the denominator equal to zero, -5 is an excluded value for option E.

Conclusion

In conclusion, -5 is an excluded value for options A, C, and E. Options B and D do not have -5 as an excluded value.

Final Answer

The final answer is:

  • Option A: Yes
  • Option B: No
  • Option C: Yes
  • Option D: No
  • Option E: Yes
    Rational Expressions and Excluded Values: A Q&A Guide =====================================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about rational expressions and excluded values.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: What is an excluded value?

A: An excluded value is a value of the variable that makes the denominator of the rational expression equal to zero. When the denominator is equal to zero, the expression is undefined, and the value is said to be an excluded value.

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

A: To determine the excluded values of a rational expression, you need to set the denominator equal to zero and solve for the variable. The values that make the denominator equal to zero are the excluded values.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can a rational expression have more than one excluded value?

A: Yes, a rational expression can have more than one excluded value. For example, if the denominator is a quadratic expression, it can have two or more real roots, which are the excluded values.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: Can a rational expression be undefined if the numerator is equal to zero?

A: No, a rational expression is only undefined if the denominator is equal to zero. If the numerator is equal to zero, the expression is equal to zero, not undefined.

Q: How do I determine if a rational expression is a polynomial or not?

A: A rational expression is a polynomial if the numerator and denominator are both polynomials, and the degree of the numerator is less than or equal to the degree of the denominator.

Q: Can a rational expression have a variable in the denominator?

A: Yes, a rational expression can have a variable in the denominator. However, the variable must be raised to an even power, otherwise, the expression is undefined.

Q: How do I add or subtract rational expressions?

A: To add or subtract rational expressions, you need to find a common denominator, and then add or subtract the numerators.

Q: Can a rational expression have a negative exponent?

A: No, a rational expression cannot have a negative exponent. However, you can rewrite a negative exponent as a positive exponent by taking the reciprocal of the expression.

Conclusion

In conclusion, rational expressions and excluded values are fundamental concepts in algebra. By understanding these concepts, you can solve equations and inequalities, and simplify rational expressions.

Final Answer

The final answer is:

  • Q: What is a rational expression? A: A fraction that contains variables and/or constants in the numerator and/or denominator.
  • Q: What is an excluded value? A: A value of the variable that makes the denominator of the rational expression equal to zero.
  • Q: How do I determine the excluded values of a rational expression? A: Set the denominator equal to zero and solve for the variable.
  • Q: What is the difference between a rational expression and a rational number? A: A rational number is a number that can be expressed as the ratio of two integers, while a rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.
  • Q: Can a rational expression have more than one excluded value? A: Yes.
  • Q: How do I simplify a rational expression? A: Factor the numerator and denominator, and then cancel out any common factors.
  • Q: Can a rational expression be undefined if the numerator is equal to zero? A: No.
  • Q: How do I determine if a rational expression is a polynomial or not? A: Check if the numerator and denominator are both polynomials, and the degree of the numerator is less than or equal to the degree of the denominator.
  • Q: Can a rational expression have a variable in the denominator? A: Yes, but the variable must be raised to an even power.
  • Q: How do I add or subtract rational expressions? A: Find a common denominator, and then add or subtract the numerators.
  • Q: Can a rational expression have a negative exponent? A: No.