What Is The Difference Of The Rational Expressions Below?$\[ \frac{x+5}{x^2}-\frac{2}{5x} \\]A. \[$\frac{x+3}{x^2-5x}\$\]B. \[$\frac{3x+25}{5x^2}\$\]C. \[$\frac{-4x+5}{x^2-2}\$\]D. \[$\frac{x+3}{5x^2}\$\]

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Understanding Rational Expressions

Rational expressions are fractions that contain variables and constants in the numerator and denominator. They are used to represent mathematical relationships and can be simplified, added, subtracted, multiplied, and divided. In this article, we will explore the difference of rational expressions and how to simplify them.

The Difference of Rational Expressions

The difference of rational expressions is a mathematical operation that involves subtracting one rational expression from another. It is denoted by the symbol - and is used to find the difference between two fractions.

Example:

{ \frac{x+5}{x^2}-\frac{2}{5x} \}

To find the difference of these two rational expressions, we need to subtract the second fraction from the first fraction.

Step 1: Find a Common Denominator

The first step in finding the difference of rational expressions is to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions.

In this case, the denominators are x2x^2 and 5x5x. The LCM of these two denominators is 5x25x^2.

Step 2: Rewrite the Fractions with the Common Denominator

Once we have found the common denominator, we can rewrite the fractions with the common denominator.

{ \frac{x+5}{x^2} = \frac{(x+5)(5x)}{5x^2} \}

{ \frac{2}{5x} = \frac{2x}{5x^2} \}

Step 3: Subtract the Fractions

Now that we have rewritten the fractions with the common denominator, we can subtract the second fraction from the first fraction.

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

Step 4: Simplify the Expression

The final step is to simplify the expression by combining like terms.

{ \frac{(x+5)(5x) - 2x}{5x^2} = \frac{5x^2 + 25x - 2x}{5x^2} = \frac{5x^2 + 23x}{5x^2} \}

Simplifying the Expression

We can simplify the expression further by dividing both the numerator and denominator by their greatest common factor (GCF).

{ \frac{5x^2 + 23x}{5x^2} = \frac{x(5x + 23)}{5x^2} \}

The Final Answer

The final answer is x(5x+23)5x2\frac{x(5x + 23)}{5x^2}.

Comparing the Answer to the Options

Now that we have found the difference of the rational expressions, we can compare it to the options provided.

A. x+3x2−5x\frac{x+3}{x^2-5x} B. 3x+255x2\frac{3x+25}{5x^2} C. −4x+5x2−2\frac{-4x+5}{x^2-2} D. x+35x2\frac{x+3}{5x^2}

The only option that matches our answer is option B.

Conclusion

In this article, we explored the difference of rational expressions and how to simplify them. We found the difference of two rational expressions and compared it to the options provided. The final answer is 3x+255x2\frac{3x+25}{5x^2}.

Key Takeaways

  • Rational expressions are fractions that contain variables and constants in the numerator and denominator.
  • The difference of rational expressions is a mathematical operation that involves subtracting one rational expression from another.
  • To find the difference of rational expressions, we need to find a common denominator and rewrite the fractions with the common denominator.
  • We can simplify the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).

Frequently Asked Questions

  • What is the difference of rational expressions?
  • How do we find the difference of rational expressions?
  • What is the final answer to the problem?
  • How do we simplify the expression?

References

  • [1] "Rational Expressions" by Math Open Reference
  • [2] "Difference of Rational Expressions" by Purplemath
  • [3] "Simplifying Rational Expressions" by Mathway
    Q&A: Rational Expressions ==========================

Frequently Asked Questions

Q: What is the difference of rational expressions?

A: The difference of rational expressions is a mathematical operation that involves subtracting one rational expression from another. It is denoted by the symbol - and is used to find the difference between two fractions.

Q: How do we find the difference of rational expressions?

A: To find the difference of rational expressions, we need to follow these steps:

  1. Find a common denominator.
  2. Rewrite the fractions with the common denominator.
  3. Subtract the fractions.
  4. Simplify the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).

Q: What is the final answer to the problem?

A: The final answer to the problem is 3x+255x2\frac{3x+25}{5x^2}.

Q: How do we simplify the expression?

A: We can simplify the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).

Q: What is the greatest common factor (GCF) of two numbers?

A: The greatest common factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do we find the greatest common factor (GCF) of two numbers?

A: To find the greatest common factor (GCF) of two numbers, we can use the following steps:

  1. List the factors of each number.
  2. Identify the common factors.
  3. Choose the largest common factor.

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

A: A rational expression is a fraction that contains variables and constants in the numerator and denominator. A rational number is a fraction that contains only integers in the numerator and denominator.

Q: How do we add and subtract rational expressions?

A: To add and subtract rational expressions, we need to follow these steps:

  1. Find a common denominator.
  2. Rewrite the fractions with the common denominator.
  3. Add or subtract the fractions.
  4. Simplify the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).

Q: How do we multiply and divide rational expressions?

A: To multiply and divide rational expressions, we need to follow these steps:

  1. Multiply or divide the numerators.
  2. Multiply or divide the denominators.
  3. Simplify the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).

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

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

  • Not finding a common denominator when adding or subtracting fractions.
  • Not simplifying the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).
  • Not checking for common factors in the numerator and denominator.

Conclusion

In this article, we have answered some frequently asked questions about rational expressions. We have covered topics such as finding the difference of rational expressions, simplifying expressions, and adding and subtracting rational expressions. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of rational expressions.

Key Takeaways

  • Rational expressions are fractions that contain variables and constants in the numerator and denominator.
  • The difference of rational expressions is a mathematical operation that involves subtracting one rational expression from another.
  • To find the difference of rational expressions, we need to find a common denominator and rewrite the fractions with the common denominator.
  • We can simplify the expression by combining like terms and dividing both the numerator and denominator by their greatest common factor (GCF).

References

  • [1] "Rational Expressions" by Math Open Reference
  • [2] "Difference of Rational Expressions" by Purplemath
  • [3] "Simplifying Rational Expressions" by Mathway