Mr. Knotts Found The Difference Of The Following Expression. Which Statement Is True About Mr. Knotts's Work?Given Expression: \[$\frac{x}{x^2-1}-\frac{1}{x-1}\$\]Step 1: \[$\frac{x}{(x+1)(x-1)}-\frac{1}{x-1}\$\]Step 2:

Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps in solving various problems. Mr. Knotts, a math enthusiast, was given an expression to simplify. The expression is $\frac{x}{x^2-1}-\frac{1}{x-1}$. In this article, we will analyze Mr. Knotts's work and determine which statement is true about his simplification.

The Given Expression

The given expression is $\frac{x}{x^2-1}-\frac{1}{x-1}$. This expression can be simplified using algebraic techniques.

Step 1: Factoring the Denominator

The first step in simplifying the expression is to factor the denominator of the first term. The denominator $x^2-1$ can be factored as $(x+1)(x-1)$. Therefore, the expression becomes $\frac{x}{(x+1)(x-1)}-\frac{1}{x-1}$.

Step 2: Finding a Common Denominator

To simplify the expression further, we need to find a common denominator for both terms. The common denominator is $(x+1)(x-1)$. Therefore, we can rewrite the second term as $\frac{-(x+1)}{(x+1)(x-1)}$.

Step 3: Combining the Terms

Now that we have a common denominator, we can combine the two terms. The expression becomes $\frac{x- (x+1)}{(x+1)(x-1)}$.

Step 4: Simplifying the Expression

The expression can be simplified further by combining like terms in the numerator. The numerator becomes $-1$. Therefore, the expression becomes $\frac{-1}{(x+1)(x-1)}$.

Discussion

Mr. Knotts's work involves simplifying the given expression using algebraic techniques. The expression is simplified by factoring the denominator, finding a common denominator, combining the terms, and simplifying the expression. The final simplified expression is $\frac{-1}{(x+1)(x-1)}$.

Conclusion

In conclusion, Mr. Knotts's work is a good example of how to simplify complex expressions using algebraic techniques. The expression is simplified by factoring the denominator, finding a common denominator, combining the terms, and simplifying the expression. The final simplified expression is $\frac{-1}{(x+1)(x-1)}$.

Key Takeaways

• Factoring the denominator is an important step in simplifying complex expressions.
• Finding a common denominator is crucial in combining terms.
• Simplifying the expression by combining like terms in the numerator is essential.
• Algebraic techniques are essential in simplifying complex expressions.

Final Answer

Introduction

In our previous article, we analyzed Mr. Knotts's work on simplifying the expression $\frac{x}{x^2-1}-\frac{1}{x-1}$. We simplified the expression using algebraic techniques and arrived at the final simplified expression $\frac{-1}{(x+1)(x-1)}$. In this article, we will answer some frequently asked questions related to Mr. Knotts's work.

Q&A

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to factor the denominator of the first term. The denominator $x^2-1$ can be factored as $(x+1)(x-1)$.

Q: Why is finding a common denominator important?

A: Finding a common denominator is crucial in combining terms. It allows us to add or subtract fractions with different denominators.

Q: How do we simplify the expression by combining like terms in the numerator?

A: To simplify the expression by combining like terms in the numerator, we need to combine the terms in the numerator. In this case, the numerator becomes $-1$.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{-1}{(x+1)(x-1)}$.

Q: What algebraic techniques are used to simplify the expression?

A: Algebraic techniques such as factoring the denominator, finding a common denominator, combining terms, and simplifying the expression by combining like terms in the numerator are used to simplify the expression.

Q: Why is it essential to simplify complex expressions?

A: Simplifying complex expressions is essential because it helps in solving various problems. It also makes the expression easier to understand and work with.

Q: Can you provide an example of a complex expression that can be simplified using algebraic techniques?

A: Yes, an example of a complex expression that can be simplified using algebraic techniques is $\frac{x^2+2x-3}{x^2-4}-\frac{x+1}{x-2}$.

Q: How do you simplify the expression $\frac{x^2+2x-3}{x^2-4}-\frac{x+1}{x-2}$?

A: To simplify the expression $\frac{x^2+2x-3}{x^2-4}-\frac{x+1}{x-2}$, we need to factor the denominator of the first term, find a common denominator, combine the terms, and simplify the expression by combining like terms in the numerator.

Q: What is the final simplified expression for $\frac{x^2+2x-3}{x^2-4}-\frac{x+1}{x-2}$?

A: The final simplified expression for $\frac{x^2+2x-3}{x^2-4}-\frac{x+1}{x-2}$ is $\frac{x+3}{(x+2)(x-2)}$.

Conclusion

In conclusion, Mr. Knotts's work on simplifying the expression $\frac{x}{x^2-1}-\frac{1}{x-1}$ is a good example of how to simplify complex expressions using algebraic techniques. The expression is simplified by factoring the denominator, finding a common denominator, combining the terms, and simplifying the expression by combining like terms in the numerator. We also answered some frequently asked questions related to Mr. Knotts's work.

Key Takeaways

• Factoring the denominator is an important step in simplifying complex expressions.
• Finding a common denominator is crucial in combining terms.
• Simplifying the expression by combining like terms in the numerator is essential.
• Algebraic techniques are essential in simplifying complex expressions.
• Simplifying complex expressions is essential because it helps in solving various problems.
• An example of a complex expression that can be simplified using algebraic techniques is $\frac{x^2+2x-3}{x^2-4}-\frac{x+1}{x-2}$.

Final Answer

The final answer is $\boxed{\frac{-1}{(x+1)(x-1)}}$.