Find The Sum Of The Rational Functions:1. $\frac{7x}{x+2} + \frac{4}{x-5}$2. $\frac{7x^2 - 31x + 8}{(x+2)(x-5)}$3. $\frac{-24x + 2}{(x+2)(x-5)}$4. 7 X 2 − 39 X + 8 ( X + 2 ) ( X − 5 ) \frac{7x^2 - 39x + 8}{(x+2)(x-5)} ( X + 2 ) ( X − 5 ) 7 X 2 − 39 X + 8 ​

Introduction

Rational functions are a type of mathematical function that involves a ratio of two polynomials. They are commonly used in various fields, including algebra, calculus, and engineering. In this article, we will focus on finding the sum of rational functions, which is a fundamental concept in mathematics. We will explore four different rational functions and provide step-by-step solutions to find their sum.

Rational Function 1: $\frac{7x}{x+2} + \frac{4}{x-5}$

To find the sum of this rational function, we need to first find a common denominator. The common denominator is the product of the two denominators, which is $(x+2)(x-5)$. We can rewrite each fraction with the common denominator:

$$\frac{7x}{x+2} + \frac{4}{x-5} = \frac{7x(x-5)}{(x+2)(x-5)} + \frac{4(x+2)}{(x+2)(x-5)}$$

Now, we can combine the two fractions by adding the numerators:

$$\frac{7x(x-5)}{(x+2)(x-5)} + \frac{4(x+2)}{(x+2)(x-5)} = \frac{7x(x-5) + 4(x+2)}{(x+2)(x-5)}$$

Simplifying the numerator, we get:

$$\frac{7x(x-5) + 4(x+2)}{(x+2)(x-5)} = \frac{7x^2 - 35x + 4x + 8}{(x+2)(x-5)}$$

Combining like terms, we get:

$$\frac{7x^2 - 35x + 4x + 8}{(x+2)(x-5)} = \frac{7x^2 - 31x + 8}{(x+2)(x-5)}$$

Rational Function 2: $\frac{7x^2 - 31x + 8}{(x+2)(x-5)}$

This rational function is already in its simplest form, so we don't need to simplify it further. We can simply use it as is to find the sum.

Rational Function 3: $\frac{-24x + 2}{(x+2)(x-5)}$

To find the sum of this rational function, we need to first find a common denominator. The common denominator is the product of the two denominators, which is $(x+2)(x-5)$. We can rewrite each fraction with the common denominator:

$$\frac{-24x + 2}{(x+2)(x-5)} = \frac{(-24x + 2)(x+2)}{(x+2)(x-5)}$$

Now, we can simplify the numerator:

$$\frac{(-24x + 2)(x+2)}{(x+2)(x-5)} = \frac{-24x^2 - 48x + 2x + 4}{(x+2)(x-5)}$$

Combining like terms, we get:

$$\frac{-24x^2 - 48x + 2x + 4}{(x+2)(x-5)} = \frac{-24x^2 - 46x + 4}{(x+2)(x-5)}$$

Rational Function 4: $\frac{7x^2 - 39x + 8}{(x+2)(x-5)}$

This rational function is already in its simplest form, so we don't need to simplify it further. We can simply use it as is to find the sum.

Finding the Sum

Now that we have simplified each rational function, we can find the sum by adding them together:

$$\frac{7x^2 - 31x + 8}{(x+2)(x-5)} + \frac{-24x^2 - 46x + 4}{(x+2)(x-5)} + \frac{7x^2 - 39x + 8}{(x+2)(x-5)}$$

We can combine the numerators by adding them together:

$$\frac{7x^2 - 31x + 8}{(x+2)(x-5)} + \frac{-24x^2 - 46x + 4}{(x+2)(x-5)} + \frac{7x^2 - 39x + 8}{(x+2)(x-5)} = \frac{7x^2 - 31x + 8 - 24x^2 - 46x + 4 + 7x^2 - 39x + 8}{(x+2)(x-5)}$$

Simplifying the numerator, we get:

$$\frac{7x^2 - 31x + 8 - 24x^2 - 46x + 4 + 7x^2 - 39x + 8}{(x+2)(x-5)} = \frac{-10x^2 - 116x + 20}{(x+2)(x-5)}$$

Conclusion

In this article, we have explored four different rational functions and provided step-by-step solutions to find their sum. We have used the concept of common denominators and combined like terms to simplify the numerators. The final sum is $\frac{-10x^2 - 116x + 20}{(x+2)(x-5)}$. This result can be used in various mathematical applications, including algebra, calculus, and engineering.

Future Work

In the future, we can explore more complex rational functions and provide step-by-step solutions to find their sum. We can also use this concept to solve more advanced mathematical problems, such as finding the sum of rational functions with multiple variables.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Simplifying Rational Functions" by Khan Academy
• [3] "Adding Rational Functions" by Purplemath

Glossary

• Rational Function: A type of mathematical function that involves a ratio of two polynomials.
• Common Denominator: The product of the two denominators of a rational function.
• Like Terms: Terms that have the same variable and exponent.
• Simplifying: Reducing a rational function to its simplest form by combining like terms.

Appendix

• Rational Function 1: $\frac{7x}{x+2} + \frac{4}{x-5}$
• Rational Function 2: $\frac{7x^2 - 31x + 8}{(x+2)(x-5)}$
• Rational Function 3: $\frac{-24x + 2}{(x+2)(x-5)}$
• Rational Function 4: $\frac{7x^2 - 39x + 8}{(x+2)(x-5)}$

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Q: What is a rational function?

A: A rational function is a type of mathematical function that involves a ratio of two polynomials. It is a function that can be expressed as the ratio of two polynomials, where the numerator is a polynomial and the denominator is a non-zero polynomial.

Q: What is the common denominator of a rational function?

A: The common denominator of a rational function is the product of the two denominators. It is the denominator that is common to both fractions in a rational function.

Q: How do I simplify a rational function?

A: To simplify a rational function, you need to combine like terms in the numerator and denominator. You can do this by adding or subtracting the numerators and denominators separately.

Q: What is the difference between a rational function and a polynomial?

A: A rational function is a function that involves a ratio of two polynomials, while a polynomial is a function that is expressed as a sum of terms, where each term is a product of a coefficient and a variable raised to a power.

Q: Can I add or subtract rational functions?

A: Yes, you can add or subtract rational functions by combining the numerators and denominators separately. However, you need to make sure that the denominators are the same before you can add or subtract the numerators.

Q: How do I find the sum of rational functions?

A: To find the sum of rational functions, you need to first find a common denominator. Then, you can combine the numerators and denominators separately to get the final sum.

Q: Can I multiply or divide rational functions?

A: Yes, you can multiply or divide rational functions by multiplying or dividing the numerators and denominators separately. However, you need to make sure that the denominators are the same before you can multiply or divide the numerators.

Q: What is the final sum of the rational functions in the previous article?

A: The final sum of the rational functions in the previous article is $\frac{-10x^2 - 116x + 20}{(x+2)(x-5)}$.

Q: Can I use rational functions in real-world applications?

A: Yes, rational functions can be used in various real-world applications, including algebra, calculus, and engineering. They can be used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits.

Q: What are some common mistakes to avoid when working with rational functions?

A: Some common mistakes to avoid when working with rational functions include:

• Not finding a common denominator before adding or subtracting rational functions
• Not combining like terms in the numerator and denominator
• Not checking if the denominators are the same before multiplying or dividing rational functions
• Not simplifying the rational function before using it in a real-world application

Q: How can I practice working with rational functions?

A: You can practice working with rational functions by:

• Simplifying rational functions with different denominators
• Adding or subtracting rational functions with different denominators
• Multiplying or dividing rational functions with different denominators
• Using rational functions to model real-world phenomena

Q: What are some resources for learning more about rational functions?

A: Some resources for learning more about rational functions include:

• Math Open Reference: A free online reference book that covers rational functions and other mathematical topics.
• Khan Academy: A free online learning platform that covers rational functions and other mathematical topics.
• Purplemath: A free online learning platform that covers rational functions and other mathematical topics.

Note: The above content is in markdown form and has been optimized for SEO. The article is a Q&A format and provides answers to frequently asked questions about rational functions.