Rewriting Rational Expressions With Common Denominator (5a+2)(a-4)(a+2)

#h1 Rewriting Rational Expressions with Denominator (5a+2)(a-4)(a+2)

In the realm of algebra, manipulating rational expressions is a fundamental skill. Often, we encounter situations where we need to combine or compare rational expressions, which requires them to have a common denominator. This article delves into the process of rewriting rational expressions with a specified denominator, focusing on the expressions $\frac{2}{5a^2-18a-8}$ and $\frac{8a}{5a^2+12a+4}$, with the target common denominator being $(5a+2)(a-4)(a+2)$.

#h2 Understanding Rational Expressions and Common Denominators

Before we dive into the specific problem, let's establish a solid understanding of rational expressions and common denominators.

What are Rational Expressions?

A rational expression is simply a fraction where the numerator and denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples of rational expressions include $\frac{x+1}{x-2}$, $\frac{3a^2}{a+5}$, and the expressions we'll be working with in this article.

The Importance of Common Denominators

Just like with numerical fractions, we cannot directly add or subtract rational expressions unless they share a common denominator. The common denominator acts as a unifying foundation, allowing us to combine the numerators while maintaining the integrity of the fractions. Think of it like trying to add apples and oranges – you need a common unit (like "fruit") to perform the addition meaningfully. In the case of rational expressions, the common denominator provides that common unit.

Finding the least common denominator (LCD) is often the most efficient approach. The LCD is the smallest expression that is divisible by both original denominators. Once we have the LCD, we can rewrite each rational expression with this new denominator, making addition and subtraction straightforward.

#h2 Rewriting $\frac{2}{5a^2-18a-8}$ with the Denominator $(5a+2)(a-4)(a+2)$

Our first task is to rewrite the rational expression $\frac{2}{5a^2-18a-8}$ so that its denominator matches the target denominator $(5a+2)(a-4)(a+2)$. This involves a process of factoring and multiplication.

Step 1: Factor the Original Denominator

The first step is to factor the denominator $5a^2-18a-8$. Factoring involves breaking down a polynomial into a product of simpler expressions. In this case, we need to find two binomials that multiply to give $5a^2-18a-8$. This can be done using various techniques, such as factoring by grouping or using the quadratic formula. The factored form of $5a^2-18a-8$ is $(5a+2)(a-4)$.

Step 2: Identify the Missing Factors

Now, we compare the factored original denominator $(5a+2)(a-4)$ with the target denominator $(5a+2)(a-4)(a+2)$. We can see that the original denominator is missing the factor $(a+2)$. This missing factor is crucial for rewriting the expression with the desired denominator.

Step 3: Multiply by the Missing Factor

To obtain the target denominator, we need to multiply both the numerator and the denominator of the original expression by the missing factor $(a+2)$. This is equivalent to multiplying the expression by 1, which doesn't change its value but allows us to rewrite it in a different form. So, we have:

$$\frac{2}{5a^2-18a-8} = \frac{2}{(5a+2)(a-4)} = \frac{2(a+2)}{(5a+2)(a-4)(a+2)}$$

Step 4: Simplify (Optional)

In some cases, you might be able to simplify the resulting expression by expanding the numerator. However, for the purpose of having a common denominator, this step is not strictly necessary. The rewritten expression is $\frac{2(a+2)}{(5a+2)(a-4)(a+2)}$.

#h2 Rewriting $\frac{8a}{5a^2+12a+4}$ with the Denominator $(5a+2)(a-4)(a+2)$

Next, we'll apply the same process to rewrite the rational expression $\frac{8a}{5a^2+12a+4}$ with the target denominator $(5a+2)(a-4)(a+2)$.

Step 1: Factor the Original Denominator

We begin by factoring the denominator $5a^2+12a+4$. Similar to the previous example, we need to find two binomials that multiply to give this quadratic expression. The factored form of $5a^2+12a+4$ is $(5a+2)(a+2)$.

Step 2: Identify the Missing Factors

Comparing the factored original denominator $(5a+2)(a+2)$ with the target denominator $(5a+2)(a-4)(a+2)$, we observe that the original denominator is missing the factor $(a-4)$.

Step 3: Multiply by the Missing Factor

To achieve the target denominator, we multiply both the numerator and the denominator of the original expression by the missing factor $(a-4)$:

$$\frac{8a}{5a^2+12a+4} = \frac{8a}{(5a+2)(a+2)} = \frac{8a(a-4)}{(5a+2)(a-4)(a+2)}$$

Step 4: Simplify (Optional)

Again, simplifying by expanding the numerator is optional. The rewritten expression is $\frac{8a(a-4)}{(5a+2)(a-4)(a+2)}$.

#h2 Conclusion

We have successfully rewritten both rational expressions with the common denominator $(5a+2)(a-4)(a+2)$. The rewritten expressions are:

• $\frac{2}{5a^2-18a-8} = \frac{2(a+2)}{(5a+2)(a-4)(a+2)}$
• $\frac{8a}{5a^2+12a+4} = \frac{8a(a-4)}{(5a+2)(a-4)(a+2)}$

Having these expressions with a common denominator now allows us to perform operations such as addition or subtraction, should the need arise. The process of rewriting rational expressions with a common denominator is a crucial technique in algebra, enabling us to manipulate and simplify complex expressions effectively. Understanding factoring, identifying missing factors, and multiplying by appropriate expressions are the key steps to mastering this skill.

This process of rewriting rational expressions with a common denominator is not just a mathematical exercise; it's a fundamental tool in various fields, including calculus, engineering, and physics. For instance, when solving equations involving rational functions or when dealing with partial fraction decomposition, the ability to find a common denominator is indispensable.

In summary, the ability to rewrite rational expressions with a common denominator is a vital algebraic skill. By mastering the steps of factoring, identifying missing factors, and multiplying by the appropriate expression, you can confidently manipulate and simplify rational expressions, paving the way for success in more advanced mathematical concepts and applications.

#h2 Further Exploration

To solidify your understanding, consider working through additional examples. Try different rational expressions and target denominators. Pay attention to the factoring process, as it's often the most challenging step. You can also explore online resources and textbooks for more practice problems and explanations. Remember, practice makes perfect!

Furthermore, delve into the concept of the least common denominator (LCD). While any common denominator will work, the LCD is the most efficient choice as it results in the simplest form of the rewritten expressions. Understanding how to find the LCD will save you time and effort in the long run.

Finally, consider how these skills apply to real-world problems. Look for examples in fields like physics or engineering where rational expressions are used to model relationships between variables. By seeing the practical applications of these concepts, you'll gain a deeper appreciation for their importance.