Simplifying Rational Expressions: A Step-by-Step Guide

Let's dive into simplifying the rational expression: $\frac{4 p^2+32 p}{p-2} \cdot \frac{p^2+p-6}{2 p^3+16 p^2}$. Simplifying expressions like this involves factoring, canceling common factors, and making sure we end up with the most reduced form possible. It might seem daunting at first, but trust me, once you get the hang of it, it's almost like solving a puzzle. So, grab your metaphorical magnifying glass, and let’s break it down step by step!

Step 1: Factor Everything!

The golden rule when simplifying rational expressions is to factor, factor, factor! Factoring allows us to identify common terms in the numerator and denominator that can be canceled out. Let's start with the first fraction, $\frac{4 p^2+32 p}{p-2}$. We can factor out a $4p$ from the numerator: $4p^2 + 32p = 4p(p + 8)$. The denominator, $p-2$, is already in its simplest form, so we'll leave it as is.

Now, let's move on to the second fraction, $\frac{p^2+p-6}{2 p^3+16 p^2}$. The numerator is a quadratic expression, $p^2 + p - 6$. We need to find two numbers that multiply to -6 and add up to 1. Those numbers are 3 and -2. So, we can factor the numerator as $(p + 3)(p - 2)$. For the denominator, $2p^3 + 16p^2$, we can factor out $2p^2$, giving us $2p^2(p + 8)$. Alright, now that we've factored everything, let's rewrite the expression with all the factored forms:

$\frac{4p(p+8)}{p-2} \cdot \frac{(p+3)(p-2)}{2p^2(p+8)}$

Step 2: Cancel Common Factors

This is the fun part! Now that we have our expression in factored form, we can start canceling out the common factors that appear in both the numerator and the denominator. Looking at our expression, we can see a few common factors:

• $(p - 2)$ appears in both the numerator and the denominator.
• $(p + 8)$ also appears in both the numerator and the denominator.
• We also have $p$ in the numerator and $p^2$ in the denominator, so we can cancel out one $p$ from each.

Let's go ahead and cancel these out. $\frac{4p(p+8)}{p-2} \cdot \frac{(p+3)(p-2)}{2p^2(p+8)}$ becomes $\frac{4\cancel{p}(\cancel{p+8})}{\cancel{p-2}} \cdot \frac{(p+3)(\cancel{p-2})}{2p^{\cancel{2}}(\cancel{p+8})}$. After canceling, we're left with:

$\frac{4}{1} \cdot \frac{p+3}{2p}$ which simplifies to $\frac{4(p+3)}{2p}$.

Step 3: Simplify the Remaining Expression

We're almost there! Now we just need to simplify the remaining expression, $\frac{4(p+3)}{2p}$. We can see that both the numerator and the denominator have a common factor of 2. So, let's divide both by 2: $\frac{4(p+3)}{2p} = \frac{2(p+3)}{p}$.

Thus, the simplified expression is $\frac{2(p+3)}{p}$. You can also distribute the 2 in the numerator to get $\frac{2p+6}{p}$, but $\frac{2(p+3)}{p}$ is generally considered the more simplified form. And that's it! We've successfully simplified the given rational expression.

Step 4: State Restrictions (Important!)

Before we call it a day, there's one crucial step we can't forget: stating the restrictions on the variable. Restrictions are values of $p$ that would make the original expression undefined. Remember, a rational expression is undefined when the denominator is equal to zero. So, we need to look back at the original expression and identify any values of $p$ that would make any of the denominators zero.

Our original expression was $\frac{4 p^2+32 p}{p-2} \cdot \frac{p^2+p-6}{2 p^3+16 p^2}$. Let's examine each denominator:

• $p - 2$: This denominator is zero when $p = 2$.
• $2p^3 + 16p^2 = 2p^2(p + 8)$: This denominator is zero when $p = 0$ or $p = -8$.

Therefore, the restrictions on $p$ are $p \neq 2$, $p \neq 0$, and $p \neq -8$. These values would make the original expression undefined, so we need to exclude them from the domain of the expression.

Putting It All Together

So, to recap, we started with the expression $\frac{4 p^2+32 p}{p-2} \cdot \frac{p^2+p-6}{2 p^3+16 p^2}$, and after factoring, canceling, and simplifying, we arrived at the simplified expression $\frac{2(p+3)}{p}$, with the restrictions $p \neq 2$, $p \neq 0$, and $p \neq -8$.

Why are Restrictions Important?

Restrictions are super important because they tell us the values that $p$ cannot be. If we plug $p = 2$, $p = 0$, or $p = -8$ into the original expression, we'd be dividing by zero, which is a big no-no in math land. Even though our simplified expression might look like it's defined at these points, we have to remember where we came from! The restrictions preserve the original expression's domain.

Common Mistakes to Avoid

• Forgetting to Factor Completely: Always make sure you've factored everything as much as possible before you start canceling. Missing a factor can lead to incorrect simplification.
• Canceling Terms Instead of Factors: You can only cancel factors, not individual terms. For example, you can't cancel the $p$ in $\frac{2p + 6}{p}$ unless you factor the numerator first.
• Ignoring Restrictions: This is a big one! Always, always state the restrictions on the variable. It's an essential part of the problem.
• Simplifying Too Early: Make sure you have factored fully before you start canceling. Sometimes you might be tempted to simplify early on, but you may miss some factors.

Practice Makes Perfect

The best way to get comfortable with simplifying rational expressions is to practice, practice, practice! Work through a variety of examples, and don't be afraid to make mistakes. Mistakes are a great way to learn. The more you practice, the quicker you'll become at factoring, canceling, and simplifying. You'll start seeing patterns and recognizing common factors almost instantly.

Simplifying rational expressions might seem intimidating at first, but with a systematic approach and plenty of practice, you'll be a pro in no time! Just remember to factor completely, cancel common factors, simplify the remaining expression, and, most importantly, state those restrictions! Guys, keep up the great work, and happy simplifying!

Now go forth and simplify some rational expressions, you've got this!