Determine The Two Rational Expressions Whose Difference Completes This Equation.Expressions:1. $\frac{1}{x^3+6 \pi}$2. $\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}}$3. $\frac{1}{x^2+6}$Drag Each Expression To The Correct

Introduction

Rational expressions are a fundamental concept in algebra, and understanding how to manipulate them is crucial for solving various mathematical problems. In this article, we will explore the process of determining two rational expressions whose difference completes a given equation. We will examine three different expressions and drag each expression to the correct category.

Expression 1: $\frac{1}{x^3+6 \pi}$

Simplifying the Expression

To simplify the expression $\frac{1}{x^3+6 \pi}$, we need to factor the denominator. The denominator can be factored as follows:

$$x^3+6 \pi = (x+2\sqrt[3]{\pi})(x^2-2\sqrt[3]{\pi}x+\pi)$$

However, this factorization is not helpful in simplifying the expression. Therefore, we will leave the expression as is.

Determining the Difference

To determine the difference of the expression $\frac{1}{x^3+6 \pi}$, we need to find another rational expression whose difference with this expression completes the equation. Let's assume the other expression is $\frac{1}{x^3+6 \pi}$. We can then write the difference as follows:

$$\frac{1}{x^3+6 \pi} - \frac{1}{x^3+6 \pi} = 0$$

This difference is equal to zero, which means that the two expressions are equal. Therefore, the difference of the expression $\frac{1}{x^3+6 \pi}$ is equal to itself.

Expression 2: $\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}}$

Simplifying the Expression

To simplify the expression $\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}}$, we need to multiply the numerator and denominator by the reciprocal of the denominator. This will eliminate the fraction in the denominator.

$$\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}} = \frac{(x+2)(x^2+36)}{(x^2-3)(x-2)}$$

Determining the Difference

To determine the difference of the expression $\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}}$, we need to find another rational expression whose difference with this expression completes the equation. Let's assume the other expression is $\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}}$. We can then write the difference as follows:

$$\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}} - \frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}} = 0$$

This difference is equal to zero, which means that the two expressions are equal. Therefore, the difference of the expression $\frac{\frac{x+2}{x^2-3}}{\frac{x-2}{x^2+36}}$ is equal to itself.

Expression 3: $\frac{1}{x^2+6}$

Simplifying the Expression

To simplify the expression $\frac{1}{x^2+6}$, we need to factor the denominator. The denominator can be factored as follows:

$$x^2+6 = (x+ \sqrt{6})(x- \sqrt{6})$$

However, this factorization is not helpful in simplifying the expression. Therefore, we will leave the expression as is.

Determining the Difference

To determine the difference of the expression $\frac{1}{x^2+6}$, we need to find another rational expression whose difference with this expression completes the equation. Let's assume the other expression is $\frac{1}{x^2+6}$. We can then write the difference as follows:

$$\frac{1}{x^2+6} - \frac{1}{x^2+6} = 0$$

This difference is equal to zero, which means that the two expressions are equal. Therefore, the difference of the expression $\frac{1}{x^2+6}$ is equal to itself.

Conclusion

In this article, we have explored the process of determining two rational expressions whose difference completes a given equation. We have examined three different expressions and determined that each expression is equal to itself. This means that the difference of each expression is equal to zero.

Key Takeaways

• Rational expressions are a fundamental concept in algebra.
• Understanding how to manipulate rational expressions is crucial for solving various mathematical problems.
• The difference of a rational expression is equal to zero if the expression is equal to itself.

Final Thoughts

Introduction

In our previous article, we explored the process of determining two rational expressions whose difference completes a given equation. We examined three different expressions and determined that each expression is equal to itself. In this article, we will provide a Q&A section to help you better understand the concept and provide additional examples.

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: 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: What is the difference of a rational expression?

A: The difference of a rational expression is the result of subtracting one rational expression from another.

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

A: To determine the difference of a rational expression, you need to find another rational expression whose difference with the given expression completes the equation.

Q: What if the two rational expressions are equal?

A: If the two rational expressions are equal, then the difference of the expressions is equal to zero.

Q: Can you provide an example of a rational expression whose difference completes the equation?

A: Let's consider the expression $\frac{x+2}{x^2-3}$. To determine the difference of this expression, we need to find another rational expression whose difference with this expression completes the equation. Let's assume the other expression is $\frac{x+2}{x^2-3}$. We can then write the difference as follows:

$$\frac{x+2}{x^2-3} - \frac{x+2}{x^2-3} = 0$$

This difference is equal to zero, which means that the two expressions are equal. Therefore, the difference of the expression $\frac{x+2}{x^2-3}$ is equal to itself.

Q: Can you provide another example of a rational expression whose difference completes the equation?

A: Let's consider the expression $\frac{1}{x^2+6}$. To determine the difference of this expression, we need to find another rational expression whose difference with this expression completes the equation. Let's assume the other expression is $\frac{1}{x^2+6}$. We can then write the difference as follows:

$$\frac{1}{x^2+6} - \frac{1}{x^2+6} = 0$$

This difference is equal to zero, which means that the two expressions are equal. Therefore, the difference of the expression $\frac{1}{x^2+6}$ is equal to itself.

Q: How do I know if the two rational expressions are equal?

A: To determine if the two rational expressions are equal, you need to compare the numerator and denominator of each expression. If the numerator and denominator of both expressions are equal, then the expressions are equal.

Q: Can you provide a real-world example of a rational expression whose difference completes the equation?

A: Let's consider a real-world example of a rational expression whose difference completes the equation. Suppose we have a company that produces two types of products, A and B. The profit from product A is represented by the expression $\frac{x+2}{x^2-3}$, and the profit from product B is represented by the expression $\frac{x+2}{x^2-3}$. To determine the total profit, we need to find the difference of the two expressions. Let's assume the other expression is $\frac{x+2}{x^2-3}$. We can then write the difference as follows:

$$\frac{x+2}{x^2-3} - \frac{x+2}{x^2-3} = 0$$

This difference is equal to zero, which means that the two expressions are equal. Therefore, the difference of the expression $\frac{x+2}{x^2-3}$ is equal to itself.

Conclusion

In this article, we have provided a Q&A section to help you better understand the concept of determining two rational expressions whose difference completes a given equation. We have provided examples and explanations to help you understand the concept and provide additional examples.

Key Takeaways

• Rational expressions are a fundamental concept in algebra.
• Understanding how to manipulate rational expressions is crucial for solving various mathematical problems.
• The difference of a rational expression is equal to zero if the expression is equal to itself.

Final Thoughts

Rational expressions are a powerful tool in mathematics, and understanding how to manipulate them is essential for solving various problems. By following the steps outlined in this article, you can determine the difference of a rational expression and complete the equation.