Find The Difference Of The Rationals:${ \frac{5}{x^2-49} - \frac{4x}{x+7} }$A. { \frac{4x + 5}{(x+7)(x-7)}$}$B. { \frac{4x^2 + 28x + 5}{(x+7)(x-7)}$}$C. { \frac{-4x^2 + 33x}{(x+7)(x-7)}$}$

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, with a focus on finding the difference of two rational expressions. We will use the given problem as a case study to demonstrate the steps involved in simplifying rational expressions.

The Problem

The problem asks us to find the difference of the rational expressions:

5x2−49−4xx+7{ \frac{5}{x^2-49} - \frac{4x}{x+7} }

Step 1: Factor the Denominators

To simplify the rational expressions, we need to factor the denominators. The first denominator, x2−49{x^2-49}, can be factored as a difference of squares:

x2−49=(x−7)(x+7){ x^2-49 = (x-7)(x+7) }

The second denominator, x+7{x+7}, is already factored.

Step 2: Rewrite the Rational Expressions

Now that we have factored the denominators, we can rewrite the rational expressions:

5(x−7)(x+7)−4xx+7{ \frac{5}{(x-7)(x+7)} - \frac{4x}{x+7} }

Step 3: Find a Common Denominator

To subtract the rational expressions, we need to find a common denominator. In this case, the common denominator is (x−7)(x+7){(x-7)(x+7)}.

Step 4: Rewrite the Rational Expressions with the Common Denominator

We can rewrite the rational expressions with the common denominator:

5(x−7)(x+7)−4x(x−7)(x−7)(x+7){ \frac{5}{(x-7)(x+7)} - \frac{4x(x-7)}{(x-7)(x+7)} }

Step 5: Subtract the Rational Expressions

Now that we have the rational expressions with the common denominator, we can subtract them:

5−4x(x−7)(x−7)(x+7){ \frac{5 - 4x(x-7)}{(x-7)(x+7)} }

Step 6: Simplify the Numerator

We can simplify the numerator by expanding the expression:

5−4x(x−7)=5−4x2+28x{ 5 - 4x(x-7) = 5 - 4x^2 + 28x }

Step 7: Factor the Numerator

We can factor the numerator as follows:

5−4x2+28x=−(4x2−33x−5){ 5 - 4x^2 + 28x = -(4x^2 - 33x - 5) }

Step 8: Write the Final Answer

The final answer is:

−(4x2−33x−5)(x−7)(x+7){ \frac{-(4x^2 - 33x - 5)}{(x-7)(x+7)} }

Conclusion

In this article, we have demonstrated the steps involved in simplifying rational expressions, with a focus on finding the difference of two rational expressions. We have used the given problem as a case study to illustrate the process of simplifying rational expressions. By following these steps, you can simplify rational expressions and find the difference of two rational expressions.

Answer Choices

Now that we have simplified the rational expressions, we can compare our answer with the given answer choices:

A. {\frac{4x + 5}{(x+7)(x-7)}$}$

B. {\frac{4x^2 + 28x + 5}{(x+7)(x-7)}$}$

C. {\frac{-4x^2 + 33x}{(x+7)(x-7)}$}$

Our answer, {\frac{-(4x^2 - 33x - 5)}{(x-7)(x+7)}$}$, is equivalent to answer choice C.

Final Answer

The final answer is C. {\frac{-4x^2 + 33x}{(x+7)(x-7)}$}$.

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Introduction

In our previous article, we explored the process of simplifying rational expressions, with a focus on finding the difference of two rational expressions. In this article, we will answer some common questions related to simplifying rational expressions.

Q: What is a rational expression?

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

A: How do I simplify a rational expression?

To simplify a rational expression, you need to follow these steps:

  1. Factor the denominators.
  2. Rewrite the rational expressions with the factored denominators.
  3. Find a common denominator.
  4. Rewrite the rational expressions with the common denominator.
  5. Subtract the rational expressions.
  6. Simplify the numerator.

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

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator, while a fraction is a simple ratio of two numbers.

Q: How do I find the common denominator of two rational expressions?

To find the common denominator of two rational expressions, you need to factor the denominators and find the least common multiple (LCM) of the factors.

Q: What is the least common multiple (LCM)?

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

Q: How do I simplify a rational expression with a negative exponent?

To simplify a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent by taking the reciprocal of the expression.

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

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator, while an algebraic expression is a general expression that contains variables and/or constants.

Q: How do I simplify a rational expression with a variable in the denominator?

To simplify a rational expression with a variable in the denominator, you need to factor the denominator and rewrite the expression with the factored denominator.

Q: What is the final answer to the problem of finding the difference of two rational expressions?

The final answer to the problem of finding the difference of two rational expressions is:

−(4x2−33x−5)(x−7)(x+7){ \frac{-(4x^2 - 33x - 5)}{(x-7)(x+7)} }

Conclusion

In this article, we have answered some common questions related to simplifying rational expressions. We hope that this Q&A guide has been helpful in clarifying any doubts you may have had about simplifying rational expressions.

Additional Resources

If you need additional help with simplifying rational expressions, we recommend the following resources:

  • Khan Academy: Rational Expressions
  • Mathway: Simplifying Rational Expressions
  • Wolfram Alpha: Rational Expressions

Final Answer

The final answer to the problem of finding the difference of two rational expressions is:

−(4x2−33x−5)(x−7)(x+7){ \frac{-(4x^2 - 33x - 5)}{(x-7)(x+7)} }

This is equivalent to answer choice C.