Answer Two Questions About The Following Rational Expression:$\frac{x^2 - 8x + 16}{3x - 12}$1. What Is An Equivalent Expression In Lowest Terms? $\square$2. What Values Of X X X Must We Exclude From The Domains Of The

Mar 10, 2025 by ADMIN 220 views

Answer Two Questions About the Given Rational Expression

Understanding the Rational Expression

The given rational expression is $\frac{x^2 - 8x + 16}{3x - 12}$ . To begin with, we need to understand the properties of rational expressions and how to simplify them. A rational expression is a fraction that contains variables and/or numbers in the numerator and denominator. In this case, the numerator is a quadratic expression, and the denominator is a linear expression.

Simplifying the Rational Expression

To simplify the rational expression, we need to factor both the numerator and the denominator. The numerator can be factored as a perfect square trinomial: $x^2 - 8x + 16 = (x - 4)^2$ . The denominator can be factored as a common factor: $3x - 12 = 3(x - 4)$ . Therefore, the simplified rational expression is $\frac{(x - 4)^2}{3(x - 4)}$ .

Equivalent Expression in Lowest Terms

To find an equivalent expression in lowest terms, we need to cancel out any common factors between the numerator and the denominator. In this case, we can cancel out the factor $(x - 4)$ from both the numerator and the denominator. However, we need to be careful not to cancel out any factors that would make the denominator equal to zero. Therefore, we need to exclude $x = 4$ from the domain of the expression.

After canceling out the common factor, we are left with $\frac{x - 4}{3}$ . This is the equivalent expression in lowest terms.

Values to Exclude from the Domain

As mentioned earlier, we need to exclude $x = 4$ from the domain of the expression because it would make the denominator equal to zero. Therefore, the domain of the expression is all real numbers except $x = 4$ .

Final Answer

To summarize, the equivalent expression in lowest terms is $\frac{x - 4}{3}$ , and the values to exclude from the domain are $x = 4$ .

Conclusion

In conclusion, we have simplified the given rational expression and found an equivalent expression in lowest terms. We have also identified the values to exclude from the domain of the expression. This type of problem is an important concept in algebra and is used to solve a wide range of problems in mathematics and other fields.

Frequently Asked Questions

Q: What is a rational expression?

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

Q: How do you simplify a rational expression?

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

Q: What values do you need to exclude from the domain of a rational expression?

A: You need to exclude any values of the variable that would make the denominator equal to zero.

Q: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps to make the expression easier to work with and understand. It also helps to identify any values that need to be excluded from the domain of the expression.

Step-by-Step Solution

Step 1: Factor the numerator and denominator

The numerator can be factored as a perfect square trinomial: $x^2 - 8x + 16 = (x - 4)^2$ . The denominator can be factored as a common factor: $3x - 12 = 3(x - 4)$ .

Step 2: Cancel out common factors

We can cancel out the factor $(x - 4)$ from both the numerator and the denominator.

Step 3: Identify values to exclude from the domain

We need to exclude $x = 4$ from the domain of the expression because it would make the denominator equal to zero.

Step 4: Write the equivalent expression in lowest terms

The equivalent expression in lowest terms is $\frac{x - 4}{3}$ .

Common Mistakes to Avoid

Mistake 1: Not factoring the numerator and denominator

Not factoring the numerator and denominator can make it difficult to simplify the rational expression and identify any values that need to be excluded from the domain.

Mistake 2: Canceling out factors that would make the denominator equal to zero

Canceling out factors that would make the denominator equal to zero can result in an expression that is not defined for certain values of the variable.

Mistake 3: Not checking the domain of the expression

Not checking the domain of the expression can result in an expression that is not defined for certain values of the variable.

Real-World Applications

Rational expressions are used in a wide range of real-world applications, including:

Algebra and calculus
Physics and engineering
Economics and finance
Computer science and programming

Final Thoughts

In conclusion, simplifying rational expressions is an important concept in algebra and is used to solve a wide range of problems in mathematics and other fields. By following the steps outlined in this article, you can simplify rational expressions and identify any values that need to be excluded from the domain of the expression.

Frequently Asked Questions About Rational Expressions

Q: What is a rational expression?

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

Q: How do you simplify a rational expression?

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

Q: What values do you need to exclude from the domain of a rational expression?

A: You need to exclude any values of the variable that would make the denominator equal to zero.

Q: Why is it important to simplify rational expressions?

Q: Can you give an example of a rational expression?

A: Yes, here is an example of a rational expression: $\frac{x^2 - 8x + 16}{3x - 12}$ .

Q: How do you factor a rational expression?

A: To factor a rational expression, you need to factor both the numerator and the denominator. You can use techniques such as factoring by grouping, factoring out a greatest common factor, or using the quadratic formula.

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, such as $\frac{1}{2}$ or $\frac{3}{4}$ . A rational expression is a fraction that contains variables and/or numbers in the numerator and denominator.

Q: Can you give an example of a rational expression that is not in lowest terms?

A: Yes, here is an example of a rational expression that is not in lowest terms: $\frac{2(x - 4)}{3(x - 4)}$ . This expression can be simplified by canceling out the common factor $(x - 4)$ .

Q: How do you determine the domain of a rational expression?

A: To determine the domain of a rational expression, you need to identify any values of the variable that would make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for the variable.

Q: Can you give an example of a rational expression with a restricted domain?

A: Yes, here is an example of a rational expression with a restricted domain: $\frac{x - 4}{x - 4}$ . This expression is not defined for $x = 4$ because it would make the denominator equal to zero.

Q: How do you add or subtract rational expressions?

A: To add or subtract rational expressions, you need to have a common denominator. You can then add or subtract the numerators and keep the same denominator.

Q: Can you give an example of adding rational expressions?

A: Yes, here is an example of adding rational expressions: $\frac{x - 4}{x - 4} + \frac{2}{x - 4}$ . This expression can be simplified by combining the numerators.

Q: How do you multiply rational expressions?

A: To multiply rational expressions, you need to multiply the numerators and multiply the denominators.

Q: Can you give an example of multiplying rational expressions?

A: Yes, here is an example of multiplying rational expressions: $\frac{x - 4}{x - 4} \cdot \frac{2}{x - 4}$ . This expression can be simplified by canceling out the common factors.

Q: How do you divide rational expressions?

A: To divide rational expressions, you need to invert the second expression and multiply.

Q: Can you give an example of dividing rational expressions?

A: Yes, here is an example of dividing rational expressions: $\frac{x - 4}{x - 4} \div \frac{2}{x - 4}$ . This expression can be simplified by canceling out the common factors.