What Is The Difference?$\frac{2x+5}{x^2-3x} - \frac{3x+5}{x^3-9x} - \frac{x+1}{x^2-9}$

Introduction

When dealing with algebraic expressions, it's not uncommon to encounter complex fractions that require simplification. In this article, we will delve into the world of algebra and explore the process of simplifying a given expression involving fractions. The expression in question is $\frac{2x+5}{x^2-3x} - \frac{3x+5}{x^3-9x} - \frac{x+1}{x^2-9}$. Our goal is to simplify this expression and understand the underlying mathematical concepts that govern its behavior.

Understanding the Expression

To begin, let's examine the given expression and identify its components. We have three fractions, each with a numerator and a denominator. The first fraction is $\frac{2x+5}{x^2-3x}$, the second fraction is $\frac{3x+5}{x^3-9x}$, and the third fraction is $\frac{x+1}{x^2-9}$. Our task is to simplify this expression by combining the fractions and eliminating any common factors.

Simplifying the Expression

To simplify the expression, we need to find a common denominator for all three fractions. The common denominator is the least common multiple (LCM) of the denominators of the individual fractions. In this case, the LCM of $x^2-3x$, $x^3-9x$, and $x^2-9$ is $x^3-9x$. Once we have the common denominator, we can rewrite each fraction with the common denominator and then combine the fractions.

Finding the Common Denominator

To find the common denominator, we need to factor each denominator and identify the common factors. The first denominator is $x^2-3x$, which can be factored as $x(x-3)$. The second denominator is $x^3-9x$, which can be factored as $x(x^2-9)$ or $x(x-3)(x+3)$. The third denominator is $x^2-9$, which can be factored as $(x-3)(x+3)$. By examining the factors, we can see that the common factors are $x$ and $(x-3)$.

Rewriting the Fractions

Now that we have identified the common factors, we can rewrite each fraction with the common denominator. The first fraction becomes $\frac{(2x+5)(x+3)}{x(x-3)(x+3)}$, the second fraction becomes $\frac{(3x+5)(x-3)}{x(x-3)(x+3)(x-3)}$, and the third fraction becomes $\frac{(x+1)(x-3)}{(x-3)(x+3)}$.

Combining the Fractions

Now that we have rewritten each fraction with the common denominator, we can combine the fractions by adding or subtracting the numerators. In this case, we need to subtract the second fraction from the first fraction and then subtract the third fraction from the result.

Simplifying the Result

After combining the fractions, we need to simplify the result by eliminating any common factors. The resulting expression is $\frac{(2x+5)(x+3)-(3x+5)(x-3)-(x+1)(x-3)}{x(x-3)(x+3)}$. To simplify this expression, we need to expand the numerators and combine like terms.

Expanding the Numerators

To expand the numerators, we need to multiply each term in the numerator by the corresponding term in the denominator. The first numerator becomes $(2x+5)(x+3) = 2x^2+11x+15$, the second numerator becomes $(3x+5)(x-3) = 3x^2-9x+5$, and the third numerator becomes $(x+1)(x-3) = x^2-2x-3$.

Combining Like Terms

Now that we have expanded the numerators, we can combine like terms by adding or subtracting the coefficients of the same variables. The resulting expression is $\frac{2x^2+11x+15-3x^2+9x-5-x^2+2x+3}{x(x-3)(x+3)}$.

Simplifying the Expression

After combining like terms, we can simplify the expression by eliminating any common factors. The resulting expression is $\frac{-x^2+22x+13}{x(x-3)(x+3)}$.

Conclusion

In conclusion, we have successfully simplified the given expression involving fractions. By finding the common denominator, rewriting the fractions, combining the fractions, and simplifying the result, we have arrived at the final expression $\frac{-x^2+22x+13}{x(x-3)(x+3)}$. This expression represents the simplified form of the original expression, and it provides valuable insights into the underlying mathematical concepts that govern its behavior.

Final Answer

The final answer is $\boxed{\frac{-x^2+22x+13}{x(x-3)(x+3)}}$.

Introduction

In our previous article, we explored the process of simplifying a complex expression involving fractions. The expression in question was $\frac{2x+5}{x^2-3x} - \frac{3x+5}{x^3-9x} - \frac{x+1}{x^2-9}$. We successfully simplified the expression and arrived at the final answer $\frac{-x^2+22x+13}{x(x-3)(x+3)}$. In this article, we will address some common questions and concerns related to the simplification process.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the denominators of the individual fractions. In this case, the LCM of $x^2-3x$, $x^3-9x$, and $x^2-9$ is $x^3-9x$.

Q: How do I find the common denominator?

A: To find the common denominator, you need to factor each denominator and identify the common factors. In this case, the first denominator is $x^2-3x$, which can be factored as $x(x-3)$. The second denominator is $x^3-9x$, which can be factored as $x(x^2-9)$ or $x(x-3)(x+3)$. The third denominator is $x^2-9$, which can be factored as $(x-3)(x+3)$. By examining the factors, you can see that the common factors are $x$ and $(x-3)$.

Q: How do I rewrite the fractions with the common denominator?

A: To rewrite the fractions with the common denominator, you need to multiply each fraction by the necessary factors to obtain the common denominator. In this case, the first fraction becomes $\frac{(2x+5)(x+3)}{x(x-3)(x+3)}$, the second fraction becomes $\frac{(3x+5)(x-3)}{x(x-3)(x+3)(x-3)}$, and the third fraction becomes $\frac{(x+1)(x-3)}{(x-3)(x+3)}$.

Q: How do I combine the fractions?

A: To combine the fractions, you need to add or subtract the numerators. In this case, you need to subtract the second fraction from the first fraction and then subtract the third fraction from the result.

Q: How do I simplify the result?

A: To simplify the result, you need to eliminate any common factors. In this case, the resulting expression is $\frac{-x^2+22x+13}{x(x-3)(x+3)}$.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{-x^2+22x+13}{x(x-3)(x+3)}}$.

Common Mistakes

When simplifying complex expressions involving fractions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not finding the common denominator: Failing to find the common denominator can lead to incorrect simplification.
• Not rewriting the fractions with the common denominator: Failing to rewrite the fractions with the common denominator can lead to incorrect simplification.
• Not combining the fractions correctly: Failing to combine the fractions correctly can lead to incorrect simplification.
• Not simplifying the result: Failing to simplify the result can lead to incorrect simplification.

Conclusion

In conclusion, simplifying complex expressions involving fractions requires careful attention to detail and a thorough understanding of the underlying mathematical concepts. By following the steps outlined in this article, you can successfully simplify complex expressions and arrive at the final answer.

Final Tips

Here are some final tips to keep in mind when simplifying complex expressions involving fractions:

• Read the problem carefully: Make sure you understand the problem and what is being asked.
• Find the common denominator: Take the time to find the common denominator and rewrite the fractions accordingly.
• Combine the fractions correctly: Make sure to combine the fractions correctly and simplify the result.
• Check your work: Double-check your work to ensure that you have arrived at the correct answer.

By following these tips and avoiding common mistakes, you can successfully simplify complex expressions involving fractions and arrive at the final answer.