Simplifying (p^2-p-6)/(p^2-64) + (p^2-p-6)/(64-p^2)

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Hey guys! Let's break down how to simplify this algebraic expression. This problem involves adding two rational expressions, but there's a sneaky twist that makes it simpler than it looks. We'll go through each step, so you can totally nail similar problems in the future. Let's dive in!

Understanding the Problem

The problem we're tackling is:

(p^2 - p - 6) / (p^2 - 64) + (p^2 - p - 6) / (64 - p^2)

At first glance, it looks like we need to find a common denominator and go through the usual process of adding fractions. But, hold on! Notice anything interesting about the denominators? The denominators, (p^2 - 64) and (64 - p^2), are almost the same, just with the signs flipped. This is a crucial observation that will simplify our work.

Identifying Key Components

Before we jump into the solution, let's pinpoint the key components of this problem:

  • Rational Expressions: We're dealing with fractions where the numerator and denominator are polynomials.
  • Polynomials: Expressions like p^2 - p - 6, p^2 - 64, and 64 - p^2.
  • Factoring: We'll need to factor polynomials to simplify the expression.
  • Common Denominator: The usual way to add fractions, but we’ll find a shortcut here!

Step-by-Step Solution

Here's how we can simplify the given expression step-by-step:

1. Recognizing the Relationship Between Denominators

As we noticed earlier, the denominators are almost identical. We can rewrite the second denominator to match the first by factoring out a -1:

64 - p^2 = -1 * (p^2 - 64)

This is a super important step! It allows us to see the relationship between the two fractions more clearly. By recognizing this, we're setting ourselves up for a much easier simplification process. Factoring out the -1 is like finding a hidden key to unlock the solution.

2. Rewriting the Expression

Now, let's rewrite the original expression using this new understanding:

(p^2 - p - 6) / (p^2 - 64) + (p^2 - p - 6) / [ -1 * (p^2 - 64) ]

We can further simplify this by moving the -1 to the numerator of the second fraction:

(p^2 - p - 6) / (p^2 - 64) - (p^2 - p - 6) / (p^2 - 64)

See what we did there? By factoring out the -1, we transformed the addition problem into a subtraction problem with a common denominator. This is a classic algebraic trick that can save you a lot of time and effort.

3. Combining the Fractions

Now that we have a common denominator, we can combine the fractions. This is the part where things get really interesting! Since the numerators are identical and we are subtracting, we get:

[(p^2 - p - 6) - (p^2 - p - 6)] / (p^2 - 64)

What happens when you subtract something from itself? You get zero! So, the numerator simplifies to:

0 / (p^2 - 64)

4. Simplifying to the Final Answer

Any fraction with a numerator of 0 (and a non-zero denominator) is equal to 0. Therefore, the simplified expression is:

0

That's it! The entire expression simplifies to 0. Who would have guessed? This highlights the importance of looking for those clever shortcuts in math problems. Sometimes, the most complex-looking problems have surprisingly simple solutions.

Key Concepts Used

Let's recap the key concepts we used to solve this problem. Understanding these concepts will help you tackle similar problems with confidence.

1. Factoring

Factoring is a fundamental skill in algebra. In this problem, we factored out a -1 from the denominator. Factoring helps us simplify expressions, identify common factors, and make complex equations easier to solve. Practice factoring different types of polynomials, and you'll become a pro at spotting opportunities to simplify.

2. Recognizing Differences of Squares

While we didn't explicitly use the difference of squares factorization (a^2 - b^2 = (a + b)(a - b)), it’s good to be aware of it. The denominator p^2 - 64 fits this pattern, and in other problems, recognizing this pattern can be crucial for simplification. Keep an eye out for expressions that fit the difference of squares, as they often lead to elegant simplifications.

3. Common Denominators

Adding or subtracting fractions requires a common denominator. We used this concept implicitly when we rewrote the second fraction to have the same denominator as the first. Understanding how to find and create common denominators is essential for working with rational expressions. Remember, the goal is to make the denominators the same so you can combine the numerators.

4. The Zero Property

Any fraction with a numerator of 0 is equal to 0. This is a basic but powerful property that we used to arrive at our final answer. Keep this property in mind whenever you're simplifying expressions, as it can often lead to quick solutions.

Common Mistakes to Avoid

Even with a clear understanding of the concepts, it's easy to make mistakes. Here are some common pitfalls to watch out for:

1. Not Recognizing the Relationship Between Denominators

The biggest mistake you could make is to miss the relationship between (p^2 - 64) and (64 - p^2). Always take a moment to see if there's a simple way to make the denominators the same before jumping into more complicated methods. This can save you a ton of time and effort.

2. Incorrectly Factoring

Factoring is a critical skill, and a mistake here can derail the entire solution. Double-check your factoring to make sure it's correct. Practice factoring regularly to improve your accuracy and speed.

3. Arithmetic Errors

Simple arithmetic errors can happen, especially when dealing with negative signs. Take your time and double-check your calculations. It's easy to make a small mistake that throws off the final answer, so be diligent with your arithmetic.

4. Forgetting the Zero Property

Sometimes, in the heat of solving a problem, you might forget that 0 divided by anything is 0. Keep the zero property in mind, as it's a common shortcut in simplification problems.

Practice Problems

To really master this type of problem, practice is key! Here are a couple of similar problems you can try:

  1. Simplify: (x^2 - 4) / (x^2 - 9) + (x^2 - 4) / (9 - x^2)
  2. Simplify: (a^2 - b^2) / (a - b) - (b^2 - a^2) / (b - a)

Work through these problems step-by-step, using the techniques we discussed. Don't be afraid to make mistakes—they're a natural part of the learning process. The more you practice, the more confident you'll become in your algebraic skills.

Conclusion

Simplifying the expression (p^2 - p - 6) / (p^2 - 64) + (p^2 - p - 6) / (64 - p^2) turns out to be much easier than it initially looks. By recognizing the relationship between the denominators, we were able to simplify the expression to 0. Remember to always look for clever shortcuts and simplifications before diving into the more complex steps.

Keep practicing these types of problems, and you'll become a math whiz in no time! You've got this!