Multiplying Rational Expressions: A Step-by-Step Guide

In this comprehensive guide, we'll walk you through the process of multiplying and simplifying rational expressions. Rational expressions, which are essentially fractions involving polynomials, might seem daunting at first. But with a systematic approach, you can master the art of multiplying them. We'll use the example expression $\frac{6 x^2+30 x}{6 x^2-18 x} \cdot \frac{2 x-6}{x^2-25}$ to illustrate each step. So, let's dive in and conquer these expressions!

1. Factoring: The Key to Simplification

The first crucial step in multiplying rational expressions is factoring. Factoring breaks down polynomials into simpler components, making it easier to identify common factors that can be canceled out later. This simplifies the entire process and prevents you from working with unwieldy expressions. Look for the greatest common factor (GCF) in each polynomial and factor it out. Also, be on the lookout for special patterns like the difference of squares or perfect square trinomials, as these have specific factoring formulas. Factoring correctly is fundamental to solving these problems efficiently. By meticulously factoring each polynomial, you set the stage for simplifying the expression and arriving at the correct answer. Let's apply this to our example:

• Numerator of the first fraction: $6x^2 + 30x$. The GCF here is $6x$. Factoring it out, we get $6x(x + 5)$.
• Denominator of the first fraction: $6x^2 - 18x$. The GCF is $6x$. Factoring it out, we get $6x(x - 3)$.
• Numerator of the second fraction: $2x - 6$. The GCF is $2$. Factoring it out, we get $2(x - 3)$.
• Denominator of the second fraction: $x^2 - 25$. This is a difference of squares, which factors into $(x + 5)(x - 5)$.

Now, our expression looks like this:

$\frac{6x(x + 5)}{6x(x - 3)} \cdot \frac{2(x - 3)}{(x + 5)(x - 5)}$

2. Multiplication: Combining the Fractions

Once you've factored each polynomial, the next step is to multiply the numerators together and the denominators together. This is similar to how you multiply regular fractions. You simply combine the factored terms into a single fraction. This step sets up the simplification process by bringing all the factors into one place. Carefully multiplying across ensures that you have all the components correctly positioned for the subsequent cancellation. This combination makes it visually clearer which factors can be simplified in the next step. Remember, accuracy in this step is essential for proceeding smoothly toward the final simplified expression. For our example, we multiply the factored numerators and denominators:

$\frac{6x(x + 5) imes 2(x - 3)}{6x(x - 3) imes (x + 5)(x - 5)}$

This simplifies to:

$\frac{12x(x + 5)(x - 3)}{6x(x - 3)(x + 5)(x - 5)}$

3. Simplification: Canceling Common Factors

After multiplying, the most exciting part comes: simplification! This involves canceling out common factors that appear in both the numerator and the denominator. Think of it as dividing both the numerator and denominator by the same value, which doesn't change the overall value of the expression. This step is where the initial factoring truly pays off, as it allows you to reduce the expression to its simplest form. Scrutinize the numerator and denominator for matching factors. Canceling these factors makes the expression more manageable and reveals the final, simplified result. It's like peeling away the layers to reveal the core, uncluttered expression. Let's simplify our expression by canceling common factors:

We have $6x$, $(x + 5)$, and $(x - 3)$ in both the numerator and the denominator. Canceling these out, we get:

$\frac{12x(x + 5)(x - 3)}{6x(x - 3)(x + 5)(x - 5)} = \frac{12x}{6x} \cdot \frac{(x + 5)}{(x + 5)} \cdot \frac{(x - 3)}{(x - 3)} \cdot \frac{1}{(x - 5)}$

Which simplifies to:

$\frac{2}{x - 5}$

4. Restrictions: Identifying Excluded Values

While simplifying is essential, it's equally important to consider restrictions. Restrictions are values that would make the original expression undefined, usually because they would result in division by zero. Before simplifying, identify these values by setting each denominator factor in the original expression equal to zero and solving for $x$. These values must be excluded from the domain of the expression. Identifying restrictions ensures that your simplified expression is not only mathematically correct but also valid for all permissible values of $x$. This step adds a layer of precision to your solution, guaranteeing its accuracy and completeness. In our example, we need to consider the denominators from the original expression:

• $6x^2 - 18x = 6x(x - 3)$. Setting this to zero gives $x = 0$ and $x = 3$.
• $x^2 - 25 = (x + 5)(x - 5)$. Setting this to zero gives $x = -5$ and $x = 5$.

Therefore, the restrictions are $x \neq 0$, $x \neq 3$, $x \neq -5$, and $x \neq 5$.

5. The Final Answer: Simplified Expression with Restrictions

Finally, we present the simplified expression along with its restrictions. The simplified expression is the result of the multiplication and cancellation steps, while the restrictions are the values that $x$ cannot be. Presenting both ensures a complete and accurate answer. This final step ties everything together, providing a clear and comprehensive solution. It demonstrates that you not only know how to simplify the expression but also understand the importance of the domain and excluded values. For our example, the final answer is:

$\frac{2}{x - 5}$, where $x \neq 0$, $x \neq 3$, $x \neq -5$, and $x \neq 5$.

Conclusion

Multiplying rational expressions involves a series of steps: factoring, multiplying, simplifying, and identifying restrictions. By following these steps systematically, you can confidently tackle even the most complex expressions. Remember, factoring is the cornerstone of this process, and paying attention to restrictions ensures the accuracy and validity of your solution. Keep practicing, and you'll become a pro at multiplying rational expressions! For further learning on rational expressions, consider exploring resources like Khan Academy's Algebra I section.