Multiplying And Simplifying Rational Expressions A Step-by-Step Guide

October 16, 2025 by Scholario Team 70 views

Hey guys! Today, we're going to dive into the world of rational expressions and tackle a common problem: multiplying and simplifying them. Specifically, we'll break down how to solve the expression $\frac{x-4}{x^2-2 x-15} \cdot \frac{x-5}{x-3}$ . Don't worry if it looks intimidating at first. We'll go through each step together, making it super clear and easy to follow. By the end of this guide, you'll be a pro at handling these types of problems!

Understanding Rational Expressions

Before we jump into the multiplication and simplification, let's make sure we're all on the same page about what rational expressions are. Rational expressions are essentially fractions where the numerator and the denominator are polynomials. Think of them as the algebraic cousins of regular numerical fractions. For example, $\frac{x-4}{x^2-2 x-15}$ and $\frac{x-5}{x-3}$ are both rational expressions.

The key thing to remember is that we can perform operations like addition, subtraction, multiplication, and division on these expressions, just like we do with regular fractions. The goal is often to simplify the result into its simplest form. This usually involves factoring polynomials and canceling out common factors, which we'll see in action shortly.

When dealing with rational expressions, there's a crucial concept to keep in mind: excluded values. These are values of the variable (in this case, x) that would make the denominator of any fraction equal to zero. Why do we care? Because division by zero is undefined in mathematics. So, we need to identify these values and exclude them from our solution set. This ensures that our expressions remain mathematically valid. Factoring the denominators helps us find these excluded values easily.

In the example we’re going to solve, identifying excluded values will be a crucial first step. This will ensure that our final simplified expression is not only mathematically correct but also accounts for any restrictions on the variable x. Keep this in mind as we move forward; it’s a fundamental part of working with rational expressions.

Step 1: Factoring the Polynomials

The first step in multiplying and simplifying rational expressions is to factor all the polynomials in both the numerators and the denominators. Factoring breaks down complex expressions into simpler components, making it easier to identify common factors that can be canceled out. This is where our algebra skills come into play!

Looking at our problem, $\frac{x-4}{x^2-2 x-15} \cdot \frac{x-5}{x-3}$ , we need to factor the quadratic expression in the first denominator, which is $x^2 - 2x - 15$ . To factor this quadratic, we're looking for two numbers that multiply to -15 and add up to -2. Can you think of what they might be? Those numbers are -5 and +3. Therefore, we can factor $x^2 - 2x - 15$ as $(x - 5)(x + 3)$ .

The other terms in the expression, $(x - 4)$ and $(x - 5)$ in the numerators, and $(x - 3)$ in the denominator, are already in their simplest form and cannot be factored further. These are linear expressions, and there are no common factors to extract. Factoring the quadratic expression is the key to simplifying this problem effectively.

So, after factoring, our expression now looks like this: $\frac{x-4}{(x-5)(x+3)} \cdot \frac{x-5}{x-3}$ . Notice how the factored form makes it easier to see potential cancellations in the next step. Factoring is not just a preliminary step; it’s the foundation for simplifying rational expressions. Always make sure to factor completely before moving on to the next steps to ensure you don’t miss any opportunities for simplification.

Step 2: Multiplying the Rational Expressions

Now that we've factored the polynomials, the next step is to multiply the rational expressions. Remember, when multiplying fractions, we simply multiply the numerators together and the denominators together. It’s a straightforward process that sets us up for simplification in the next step.

In our case, we have $\frac{x-4}{(x-5)(x+3)} \cdot \frac{x-5}{x-3}$ . To multiply these expressions, we multiply the numerators $(x - 4)$ and $(x - 5)$ , and we multiply the denominators $(x - 5)(x + 3)$ and $(x - 3)$ . This gives us a new fraction:

$\frac{(x-4)(x-5)}{(x-5)(x+3)(x-3)}$

It’s super important to write this out clearly. Don’t try to skip steps or do too much in your head, especially when you’re first learning this. Writing it out makes it much easier to see what we have and what we can simplify. We now have one single rational expression, and we’re ready to look for common factors that we can cancel out. Multiplying the expressions is a mechanical step, but it’s essential for bringing everything together into one fraction before we start simplifying.

Always double-check that you’ve multiplied the numerators and denominators correctly. A small error here can throw off the entire simplification process. So, take your time and make sure each term is accounted for. This meticulous approach will save you headaches down the road and ensure you get to the correct final answer. Now, let’s move on to the exciting part: simplifying!

Step 3: Simplifying the Expression

Alright, we've reached the heart of the matter: simplifying the rational expression. This is where we look for common factors in the numerator and the denominator that we can cancel out. Simplifying is like tidying up a messy room – we're making our expression look as clean and straightforward as possible.

Looking at our expression, $\frac{(x-4)(x-5)}{(x-5)(x+3)(x-3)}$ , do you spot any common factors? We have $(x - 5)$ in both the numerator and the denominator. This means we can cancel out $(x - 5)$ from both, as long as $x$ is not equal to 5 (remember those excluded values we talked about earlier?). Canceling out this common factor is a key step in simplifying rational expressions.

After canceling out the $(x - 5)$ terms, our expression becomes:

$\frac{x-4}{(x+3)(x-3)}$

This looks much simpler already, doesn’t it? But before we declare victory, we should check if there’s anything else we can simplify. In this case, there are no more common factors between the numerator $(x - 4)$ and the denominator $(x + 3)(x - 3)$ . So, we’ve simplified as much as we can.

It's worth noting that the denominator $(x + 3)(x - 3)$ is a difference of squares, which could be expanded to $x^2 - 9$ . However, leaving it in factored form often makes it easier to see the excluded values, and in many cases, it's perfectly acceptable as a final answer. So, our simplified expression is $\frac{x-4}{(x+3)(x-3)}$ . Remember, simplification is all about making the expression as clear and concise as possible, while still maintaining its mathematical integrity.

Step 4: Identifying Excluded Values

Before we wrap things up, there's one crucial step we absolutely cannot skip: identifying the excluded values. As we mentioned earlier, excluded values are the values of x that would make the denominator of our rational expression equal to zero. These values are not allowed because division by zero is undefined in mathematics.

To find the excluded values, we need to look at the original factored form of the denominator before we canceled out any factors. This is super important because canceling out factors can sometimes hide the excluded values if we only look at the simplified expression. In our case, the original factored denominator was $(x - 5)(x + 3)(x - 3)$ .

Now, we set each factor equal to zero and solve for x:

$x - 5 = 0$ => $x = 5$
$x + 3 = 0$ => $x = -3$
$x - 3 = 0$ => $x = 3$

So, the excluded values are $x = 5$ , $x = -3$ , and $x = 3$ . This means that our simplified expression $\frac{x-4}{(x+3)(x-3)}$ is valid for all values of x except these three. We often write this as “ $x \neq 5$ , $x \neq -3$ , and $x \neq 3$ ”.

Identifying excluded values is not just a formality; it’s a fundamental part of solving rational expressions. It ensures that our solution is mathematically sound and accounts for any restrictions on the variable. Always remember to check for excluded values, and you’ll be on your way to mastering rational expressions!

Final Answer

Okay, guys, we've reached the end of our journey! Let's recap what we've done and state our final answer clearly. We started with the expression $\frac{x-4}{x^2-2 x-15} \cdot \frac{x-5}{x-3}$ and went through a step-by-step process to multiply and simplify it.

Factored the polynomials: We factored the quadratic expression $x^2 - 2x - 15$ into $(x - 5)(x + 3)$ .
Multiplied the rational expressions: We multiplied the numerators and the denominators to get $\frac{(x-4)(x-5)}{(x-5)(x+3)(x-3)}$ .
Simplified the expression: We canceled out the common factor $(x - 5)$ to obtain $\frac{x-4}{(x+3)(x-3)}$ .
Identified excluded values: We found that the excluded values are $x = 5$ , $x = -3$ , and $x = 3$ .

Therefore, our final answer is:

$\frac{x-4}{(x+3)(x-3)}$ , where $x \neq 5$ , $x \neq -3$ , and $x \neq 3$ .

And that’s it! We've successfully multiplied and simplified the rational expression, and we’ve also made sure to account for any excluded values. Remember, the key to mastering rational expressions is to take it one step at a time, factoring carefully, simplifying systematically, and always checking for those excluded values. Keep practicing, and you’ll become a pro in no time!

Multiplying and simplifying rational expressions involves factoring the polynomials, multiplying the numerators and denominators, simplifying by canceling common factors, and identifying excluded values. By following this process, one can accurately simplify complex rational expressions. We trust this guide helps you in mastering simplifying rational algebraic expressions!