What Is The Difference?${ \frac{2x+5}{x^2-3x} - \frac{3x+5}{x^3-9x} - \frac{x+1}{x^2-9} }$A. { \frac{(x+5)(x+2)}{x^3-9x}$}$B. { \frac{(x+5)(x+4)}{x^3-9x}$}$C. { \frac{-2x+11}{x^3-12x-9}$}$D.

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in calculus and algebra. It involves breaking down intricate expressions into simpler forms, making them easier to work with and understand. In this article, we will delve into the world of algebraic simplification, focusing on a specific problem that requires a deep understanding of mathematical concepts.

The Problem

The given problem is a complex algebraic expression involving fractions and polynomial expressions. The expression is:

2x+5x2βˆ’3xβˆ’3x+5x3βˆ’9xβˆ’x+1x2βˆ’9\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 find a more manageable form.

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators of each fraction. The first denominator is x2βˆ’3xx^2-3x, which can be factored as x(xβˆ’3)x(x-3). The second denominator is x3βˆ’9xx^3-9x, which can be factored as x(xβˆ’3)(x+3)x(x-3)(x+3). The third denominator is x2βˆ’9x^2-9, which can be factored as (xβˆ’3)(x+3)(x-3)(x+3).

Step 2: Rewrite the Expression with Factored Denominators

Now that we have factored the denominators, we can rewrite the expression as:

2x+5x(xβˆ’3)βˆ’3x+5x(xβˆ’3)(x+3)βˆ’x+1(xβˆ’3)(x+3)\frac{2x+5}{x(x-3)} - \frac{3x+5}{x(x-3)(x+3)} - \frac{x+1}{(x-3)(x+3)}

Step 3: Find a Common Denominator

To combine the fractions, we need to find a common denominator. The least common multiple (LCM) of the denominators is x(xβˆ’3)(x+3)x(x-3)(x+3).

Step 4: Rewrite the Expression with a Common Denominator

Now that we have found a common denominator, we can rewrite the expression as:

(2x+5)(x+3)x(xβˆ’3)(x+3)βˆ’(3x+5)(xβˆ’3)x(xβˆ’3)(x+3)βˆ’(x+1)xx(xβˆ’3)(x+3)\frac{(2x+5)(x+3)}{x(x-3)(x+3)} - \frac{(3x+5)(x-3)}{x(x-3)(x+3)} - \frac{(x+1)x}{x(x-3)(x+3)}

Step 5: Combine the Fractions

Now that we have a common denominator, we can combine the fractions by adding or subtracting the numerators.

Step 6: Simplify the Expression

After combining the fractions, we can simplify the expression by canceling out any common factors in the numerator and denominator.

Step 7: Final Answer

After simplifying the expression, we get:

(x+5)(x+4)x3βˆ’9x\frac{(x+5)(x+4)}{x^3-9x}

This is the final answer.

Conclusion

Simplifying complex algebraic expressions requires a deep understanding of mathematical concepts, including factoring, finding common denominators, and combining fractions. By following these steps, we can break down intricate expressions into simpler forms, making them easier to work with and understand.

Discussion

The given problem is a classic example of a complex algebraic expression that requires simplification. The expression involves fractions and polynomial expressions, making it challenging to work with. However, by following the steps outlined in this article, we can simplify the expression and find a more manageable form.

Answer Key

The final answer is:

(x+5)(x+4)x3βˆ’9x\frac{(x+5)(x+4)}{x^3-9x}

This is the correct answer among the options provided.

Final Thoughts

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in calculus and algebra. By mastering this skill, we can break down intricate expressions into simpler forms, making them easier to work with and understand. This article has provided a step-by-step guide on how to simplify a complex algebraic expression, and we hope that it has been helpful in understanding this important mathematical concept.

Introduction

In our previous article, we explored the concept of simplifying complex algebraic expressions, focusing on a specific problem that required a deep understanding of mathematical concepts. We broke down the expression into simpler forms, making it easier to work with and understand. In this article, we will address some of the most frequently asked questions related to simplifying complex algebraic expressions.

Q&A

Q: What is the first step in simplifying a complex algebraic expression?

A: The first step in simplifying a complex algebraic expression is to factor the denominators of each fraction. This involves breaking down the denominators into their prime factors, which can help us find a common denominator and combine the fractions.

Q: How do I find a common denominator for multiple fractions?

A: To find a common denominator for multiple fractions, we need to find the least common multiple (LCM) of the denominators. The LCM is the smallest expression that is divisible by all the denominators.

Q: What is the difference between a common denominator and a least common multiple?

A: A common denominator is an expression that is divisible by all the denominators, while a least common multiple is the smallest expression that is divisible by all the denominators.

Q: How do I combine fractions with different denominators?

A: To combine fractions with different denominators, we need to find a common denominator and then add or subtract the numerators.

Q: What is the final step in simplifying a complex algebraic expression?

A: The final step in simplifying a complex algebraic expression is to simplify the expression by canceling out any common factors in the numerator and denominator.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables by following the same steps as before. However, you may need to use more advanced techniques, such as substitution or elimination, to simplify the expression.

Q: How do I know if an expression is already simplified?

A: An expression is already simplified if it cannot be simplified further by canceling out any common factors in the numerator and denominator.

Q: What are some common mistakes to avoid when simplifying complex algebraic expressions?

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

  • Not factoring the denominators
  • Not finding a common denominator
  • Not combining fractions correctly
  • Not simplifying the expression by canceling out common factors

Q: How can I practice simplifying complex algebraic expressions?

A: You can practice simplifying complex algebraic expressions by working through example problems, such as the one we explored in our previous article. You can also try simplifying expressions on your own and then checking your work with a calculator or a math software program.

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in calculus and algebra. By mastering this skill, we can break down intricate expressions into simpler forms, making them easier to work with and understand. This article has provided a Q&A guide on how to simplify complex algebraic expressions, and we hope that it has been helpful in understanding this important mathematical concept.

Final Thoughts

Simplifying complex algebraic expressions is a challenging but rewarding task. By following the steps outlined in this article, we can simplify even the most complex expressions and gain a deeper understanding of mathematical concepts. We hope that this article has been helpful in your journey to master the art of simplifying complex algebraic expressions.