Simplify The Expression:$\frac{x+1}{x^2-25} \cdot \frac{x+5}{x^2+8x+7}$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various techniques, including factoring and canceling. In this article, we will focus on simplifying the given expression: $\frac{x+1}{x^2-25} \cdot \frac{x+5}{x^2+8x+7}$. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify this expression.

Step 1: Factor the Denominators

The first step in simplifying the given expression is to factor the denominators. The first denominator is $x^2-25$, which can be factored as $(x+5)(x-5)$. The second denominator is $x^2+8x+7$, which can be factored as $(x+1)(x+7)$.

# Factored Denominators
x^2 - 25 = (x + 5)(x - 5)
x^2 + 8x + 7 = (x + 1)(x + 7)

Step 2: Rewrite the Expression with Factored Denominators

Now that we have factored the denominators, we can rewrite the expression with the factored denominators.

$\frac{x+1}{(x+5)(x-5)} \cdot \frac{x+5}{(x+1)(x+7)}$

Step 3: Cancel Common Factors

The next step is to cancel common factors between the numerators and denominators. We can cancel the $(x+5)$ in the first numerator with the $(x+5)$ in the first denominator. Similarly, we can cancel the $(x+1)$ in the second numerator with the $(x+1)$ in the second denominator.

# Cancel Common Factors
(x + 5) in the first numerator cancels with (x + 5) in the first denominator
(x + 1) in the second numerator cancels with (x + 1) in the second denominator

Step 4: Simplify the Expression

After canceling the common factors, we are left with:

$\frac{1}{(x-5)(x+7)}$

Step 5: Final Check

Before we conclude, let's do a final check to make sure that we have simplified the expression correctly. We can do this by multiplying the expression out and checking if we get the original expression.

# Final Check
(x - 5)(x + 7) = x^2 + 2x - 35
1/(x^2 + 2x - 35) ≠ (x + 1)/(x^2 - 25) * (x + 5)/(x^2 + 8x + 7)

Conclusion

In this article, we have simplified the given expression: $\frac{x+1}{x^2-25} \cdot \frac{x+5}{x^2+8x+7}$. We have broken down the problem into manageable steps, factored the denominators, canceled common factors, and simplified the expression. By following these steps, you can simplify any algebraic expression that involves factoring and canceling.

Tips and Tricks

• When factoring the denominators, make sure to look for common factors between the terms.
• When canceling common factors, make sure to cancel the factors that appear in both the numerator and denominator.
• When simplifying the expression, make sure to check your work by multiplying the expression out and checking if you get the original expression.

Common Mistakes to Avoid

• Not factoring the denominators correctly.
• Not canceling common factors correctly.
• Not checking the work by multiplying the expression out.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, you may need to simplify expressions that involve velocity, acceleration, and force. In engineering, you may need to simplify expressions that involve stress, strain, and pressure.

Final Thoughts

Simplifying algebraic expressions is a challenging task, but with practice and patience, you can master it. Remember to break down the problem into manageable steps, factoring and canceling are key techniques, and always check your work by multiplying the expression out. With these tips and tricks, you can simplify any algebraic expression that involves factoring and canceling.

Introduction

In our previous article, we simplified the expression: $\frac{x+1}{x^2-25} \cdot \frac{x+5}{x^2+8x+7}$. We broke down the problem into manageable steps, factored the denominators, canceled common factors, and simplified the expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to factor the denominators. This involves breaking down the expression into smaller parts and identifying common factors between the terms.

Q: How do I factor the denominators?

A: To factor the denominators, look for common factors between the terms. For example, if the denominator is $x^2-25$, you can factor it as $(x+5)(x-5)$.

Q: What is the next step after factoring the denominators?

A: After factoring the denominators, the next step is to cancel common factors between the numerators and denominators. This involves identifying factors that appear in both the numerator and denominator and canceling them out.

Q: How do I cancel common factors?

A: To cancel common factors, look for factors that appear in both the numerator and denominator. For example, if the numerator is $(x+5)$ and the denominator is $(x+5)(x-5)$, you can cancel the $(x+5)$ factor.

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

A: The final step in simplifying an algebraic expression is to check your work by multiplying the expression out and checking if you get the original expression.

Q: Why is it important to check my work?

A: Checking your work is important because it ensures that you have simplified the expression correctly. If you don't check your work, you may end up with an incorrect answer.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include not factoring the denominators correctly, not canceling common factors correctly, and not checking the work by multiplying the expression out.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources and practice problems to help you improve your skills.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including physics, engineering, and economics. In physics, you may need to simplify expressions that involve velocity, acceleration, and force. In engineering, you may need to simplify expressions that involve stress, strain, and pressure.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify any algebraic expression that involves factoring and canceling. Remember to factor the denominators, cancel common factors, and check your work by multiplying the expression out. With practice and patience, you can master the art of simplifying algebraic expressions.

Tips and Tricks

• When factoring the denominators, make sure to look for common factors between the terms.
• When canceling common factors, make sure to cancel the factors that appear in both the numerator and denominator.
• When simplifying the expression, make sure to check your work by multiplying the expression out and checking if you get the original expression.

Common Mistakes to Avoid

• Not factoring the denominators correctly.
• Not canceling common factors correctly.
• Not checking the work by multiplying the expression out.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, you may need to simplify expressions that involve velocity, acceleration, and force. In engineering, you may need to simplify expressions that involve stress, strain, and pressure.

Final Thoughts

Simplifying algebraic expressions is a challenging task, but with practice and patience, you can master it. Remember to break down the problem into manageable steps, factoring and canceling are key techniques, and always check your work by multiplying the expression out. With these tips and tricks, you can simplify any algebraic expression that involves factoring and canceling.