Simplifying Rational Expressions: A Step-by-Step Guide

Oct 23, 2025 by Dimemap Team 55 views

Hey guys! Let's dive into the world of rational expressions! Today, we're going to break down how to multiply and simplify a rational expression. It might sound a bit intimidating at first, but trust me, with a few simple steps, you'll be acing these problems in no time. We'll be working with the expression: $\frac{x^2-25}{x^2-64} \cdot \frac{x^2-3 x-40}{x^2+2 x-35}$ . Our goal is to simplify this as much as possible, which means reducing it to its lowest terms. This is a common task in algebra, and understanding it is crucial for more advanced math concepts. Ready to get started? Let's go!

Step 1: Factor, Factor, Factor!

The first and most crucial step in simplifying rational expressions is to factor everything you see. Factoring means breaking down each part of the expression (numerator and denominator) into simpler terms. Think of it like taking apart a Lego structure to see the individual bricks. We'll use different factoring techniques, depending on the type of expression. Remember that factoring is the process of finding the numbers or expressions that multiply together to give the original expression. It's like finding the building blocks of the expression. This step allows us to identify and cancel out common factors, which is the key to simplification. Without factoring, you're stuck with a complicated expression you can't reduce. So, let's look at our expression: $\frac{x^2-25}{x^2-64} \cdot \frac{x^2-3 x-40}{x^2+2 x-35}$ . Let's start with the first part, $x^2 - 25$ . This is a difference of squares. The difference of squares is a special pattern we can factor. It follows the pattern: $a^2 - b^2 = (a + b)(a - b)$ . In our case, $x^2 - 25$ can be factored as $(x + 5)(x - 5)$ . Next, we have $x^2 - 64$ , which is also a difference of squares. It factors into $(x + 8)(x - 8)$ . Now, let's move on to the second fraction. We have $x^2 - 3x - 40$ . This is a quadratic expression that we can factor by finding two numbers that multiply to -40 and add up to -3. Those numbers are -8 and 5. Therefore, $x^2 - 3x - 40$ factors into $(x - 8)(x + 5)$ . Finally, we have $x^2 + 2x - 35$ . We need two numbers that multiply to -35 and add up to 2. Those numbers are 7 and -5. So, $x^2 + 2x - 35$ factors into $(x + 7)(x - 5)$ . Now, let's rewrite the entire expression with the factored forms: $\frac{(x + 5)(x - 5)}{(x + 8)(x - 8)} \cdot \frac{(x - 8)(x + 5)}{(x + 7)(x - 5)}$ . See, factoring makes things much clearer!

Step 2: Cancel Common Factors

Alright, now that we've got everything factored, it's time to play the cancellation game! This is where the magic of simplification happens. The whole point of factoring is to identify common factors in the numerator and denominator. When a factor appears in both the top and the bottom, we can cancel them out, just like dividing a number by itself. This is based on the fundamental property of fractions: if you multiply or divide both the numerator and denominator by the same non-zero number, the value of the fraction remains the same. The cancellation process streamlines the expression and helps us see its true, simplified form. Remember, you can only cancel factors, not terms that are added or subtracted. The key here is to look for identical expressions in the numerator and the denominator. For example, if you see an $(x + 5)$ in the numerator and an $(x + 5)$ in the denominator, you can cancel them out. It's like saying, "Goodbye, you two!" Let's go back to our factored expression: $\frac{(x + 5)(x - 5)}{(x + 8)(x - 8)} \cdot \frac{(x - 8)(x + 5)}{(x + 7)(x - 5)}$ . Now, let's see what we can cancel. We can see an $(x - 5)$ in the numerator of the first fraction and an $(x - 5)$ in the denominator of the second fraction. Those cancel out! We can also see an $(x - 8)$ in the denominator of the first fraction and an $(x - 8)$ in the numerator of the second fraction. Those also cancel out! After canceling, we're left with: $\frac{(x + 5)}{(x + 8)} \cdot \frac{(x + 5)}{(x + 7)}$ .

Step 3: Multiply (If Necessary) and State Excluded Values

So, after the cancellations, we have $\frac{(x + 5)}{(x + 8)} \cdot \frac{(x + 5)}{(x + 7)}$ . Now, we multiply the remaining factors in the numerators and denominators. This is often as simple as just writing them next to each other, but let's be thorough. Multiplying the numerators gives us $(x + 5)(x + 5)$ , which can be written as $(x + 5)^2$ or $x^2 + 10x + 25$ if you expand it. Multiplying the denominators gives us $(x + 8)(x + 7)$ , which expands to $x^2 + 15x + 56$ . Therefore, the simplified expression is $\frac{(x + 5)(x + 5)}{(x + 8)(x + 7)}$ or $\frac{(x + 5)^2}{(x + 8)(x + 7)}$ or $\frac{x^2 + 10x + 25}{x^2 + 15x + 56}$ . But wait, there's one more important thing to do: state the excluded values. Excluded values are the values of x that would make the original denominator equal to zero. Remember, you can't divide by zero! To find these values, we look back at the original denominators before any canceling happened. The original denominators were $x^2 - 64$ and $x^2 + 2x - 35$ . The factors of these denominators were $(x + 8)$ , $(x - 8)$ , $(x + 7)$ , and $(x - 5)$ . So, the values that make these factors equal to zero are $x = -8$ , $x = 8$ , $x = -7$ , and $x = 5$ . Thus, our final answer is $\frac{(x + 5)^2}{(x + 8)(x + 7)}$ , where $x \neq -8, 8, -7, 5$ . And that, my friends, is how you multiply and simplify rational expressions. You did it! These excluded values are essential to note because, although they might not appear to cause an issue in the simplified form, they are restrictions based on the original expression. These values should always be stated as part of your final answer, because they indicate the values of x that would make the original expression undefined.

Summary of Steps

Let's recap the key steps involved in simplifying rational expressions:

Factor: Factor all numerators and denominators completely.
Cancel: Cancel any common factors between numerators and denominators.
Multiply (if necessary): Multiply the remaining factors to simplify the expression.
State Excluded Values: Identify and state any values of x that would make the original denominator equal to zero.

Tips and Tricks

Here are some helpful tips to make simplifying rational expressions easier:

Practice, practice, practice! The more you work with these problems, the more comfortable you'll become.
Double-check your factoring. Factoring errors are a common source of mistakes. Take your time and make sure you've factored correctly.
Always look for the difference of squares and other special factoring patterns. They can save you a lot of time.
Write down the excluded values before you start canceling. This can help you remember them.
Don't be afraid to rewrite things. Sometimes rewriting a fraction can show you the commonalities that can be simplified.
Use different colors. Sometimes using different colors while factoring or canceling can help you stay organized.
Ask for help! If you're struggling, don't hesitate to ask your teacher, classmates, or online resources for assistance.

Conclusion

Congratulations, you've successfully navigated the process of simplifying a rational expression! Remember that this skill is a building block for more advanced mathematical concepts. Keep practicing, and you'll become a pro in no time. Keep the steps in mind: Factor, Cancel, Multiply, and State Excluded Values, and you will do great. If you work at the practice, you'll find that these steps become natural. Keep up the good work, and happy simplifying, folks!