Subtracting Rational Expressions A Step-by-Step Guide
In the realm of mathematics, particularly within algebra, subtracting rational expressions is a fundamental operation. This article delves into the process of subtracting two rational expressions: $\frac{9}{2 z^2+6 z-36}-\frac{6 z+2}{2 z^2+4 z-30}$. We will break down the steps involved, ensuring a clear understanding for anyone tackling similar problems. This guide aims to provide not just the solution, but also the methodology, making it easier to approach and solve such mathematical challenges. Our primary goal is to transform the given expressions into a single rational expression, under the assumption that the denominator is never zero. This is crucial because division by zero is undefined in mathematics, and we must always consider the domain of our expressions.
Understanding Rational Expressions
Before we dive into the subtraction process, it’s essential to understand what rational expressions are. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include $x^2 + 3x - 2$ and $5z^3 - 7z + 1$. Rational expressions can appear complex, but they are governed by the same principles as regular fractions. Therefore, the rules we apply to adding, subtracting, multiplying, and dividing fractions are also applicable here.
The challenge in dealing with rational expressions often lies in the algebraic manipulation required to simplify them. This typically involves factoring polynomials, finding common denominators, and reducing expressions to their simplest forms. Mastering these techniques is crucial for success in algebra and higher-level mathematics. When subtracting rational expressions, the key step is to find a common denominator. Just as with numerical fractions, we cannot directly subtract fractions with different denominators. The common denominator allows us to combine the numerators and express the result as a single fraction. This process often involves several steps, including factoring the denominators, identifying the least common multiple (LCM), and adjusting the numerators accordingly.
Factoring the Denominators
The first critical step in subtracting rational expressions is factoring the denominators. This process is essential for identifying the least common denominator (LCD). Factoring involves breaking down a polynomial into its constituent factors, which are simpler expressions that, when multiplied together, yield the original polynomial. In our case, we have two denominators: $2z^2 + 6z - 36$ and $2z^2 + 4z - 30$. Let's factor each one individually.
For the first denominator, $2z^2 + 6z - 36$, we can start by factoring out the greatest common factor (GCF), which is 2. This gives us $2(z^2 + 3z - 18)$. Now, we need to factor the quadratic expression $z^2 + 3z - 18$. We are looking for two numbers that multiply to -18 and add to 3. These numbers are 6 and -3. Therefore, we can factor the quadratic as $(z + 6)(z - 3)$. Combining this with the GCF, the factored form of the first denominator is $2(z + 6)(z - 3)$.
Moving on to the second denominator, $2z^2 + 4z - 30$, we again start by factoring out the GCF, which is also 2. This gives us $2(z^2 + 2z - 15)$. Now, we factor the quadratic expression $z^2 + 2z - 15$. We need two numbers that multiply to -15 and add to 2. These numbers are 5 and -3. Thus, the quadratic factors as $(z + 5)(z - 3)$. Including the GCF, the factored form of the second denominator is $2(z + 5)(z - 3)$. Factoring the denominators is a crucial step because it allows us to identify the common and unique factors, which are necessary for determining the least common denominator.
Finding the Least Common Denominator (LCD)
Once we have factored the denominators, the next step is to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. To find the LCD, we consider all the unique factors present in the denominators and take the highest power of each factor.
Our factored denominators are $2(z + 6)(z - 3)$ and $2(z + 5)(z - 3)$. The unique factors are 2, $(z + 6)$, $(z - 3)$, and $(z + 5)$. The LCD is the product of these factors, each raised to the highest power it appears in any of the denominators. In this case, each factor appears only once in each denominator, so the LCD is $2(z + 6)(z - 3)(z + 5)$. The LCD is a crucial element in subtracting rational expressions because it allows us to rewrite each fraction with a common denominator, making the subtraction process straightforward. Without a common denominator, we cannot directly combine the numerators.
Rewriting the Fractions with the LCD
After identifying the LCD, the next step is to rewrite each fraction with the LCD as its denominator. This involves multiplying the numerator and denominator of each fraction by the factors that are missing from its original denominator. This process ensures that the value of the fraction remains unchanged while allowing us to combine them easily.
For the first fraction, $\frac{9}{2(z + 6)(z - 3)}$, the LCD is $2(z + 6)(z - 3)(z + 5)$. The missing factor in the denominator is $(z + 5)$. Therefore, we multiply both the numerator and the denominator by $(z + 5)$, resulting in $\frac{9(z + 5)}{2(z + 6)(z - 3)(z + 5)}$.
For the second fraction, $\frac{6z + 2}{2(z + 5)(z - 3)}$, the LCD is the same, $2(z + 6)(z - 3)(z + 5)$. The missing factor in the denominator is $(z + 6)$. So, we multiply both the numerator and the denominator by $(z + 6)$, giving us $\frac{(6z + 2)(z + 6)}{2(z + 5)(z - 3)(z + 6)}$. Rewriting the fractions with the LCD is a critical step because it sets the stage for combining the numerators. Once both fractions have the same denominator, we can subtract the numerators and simplify the resulting expression.
Subtracting the Numerators
Now that both fractions have the same denominator, we can subtract the numerators. This step involves combining the numerators while keeping the common denominator. It’s crucial to pay close attention to the signs and distribute any negative signs properly.
We have the expressions $\frac{9(z + 5)}{2(z + 6)(z - 3)(z + 5)}$ and $\frac{(6z + 2)(z + 6)}{2(z + 5)(z - 3)(z + 6)}$. Subtracting the second fraction from the first, we get:
Next, we need to expand the numerators. For the first term, $9(z + 5)$, we distribute the 9 to get $9z + 45$. For the second term, $(6z + 2)(z + 6)$, we use the distributive property (also known as FOIL) to expand the product:
Now, we substitute these expansions back into the expression:
Distribute the negative sign to the second term in the numerator:
Combine like terms in the numerator:
Subtracting the numerators is a crucial step that requires careful attention to detail. The correct application of the distributive property and combining like terms are essential for obtaining the correct result.
Simplifying the Resulting Expression
After subtracting the numerators, the final step is to simplify the resulting expression. This involves factoring the numerator, if possible, and canceling any common factors between the numerator and the denominator. Simplifying the expression ensures that we present the answer in its most concise and understandable form.
Our current expression is:
First, let's try to factor the numerator, $-6z^2 - 29z + 33$. Factoring a quadratic expression can be challenging, but we can use various techniques, such as the quadratic formula or factoring by grouping. In this case, we are looking for two binomials that multiply to give the quadratic expression. After some trial and error, we find that:
So, our expression becomes:
Now, we look for common factors between the numerator and the denominator. In this case, there are no common factors that can be canceled. The expression is now in its simplest form. Simplifying the expression is a critical step because it ensures that we present the answer in its most concise and understandable form. By factoring and canceling common factors, we reduce the expression to its lowest terms.
Final Answer
After performing all the steps, the final simplified expression is:
This is the result of subtracting the given rational expressions. We have successfully combined the two expressions into a single rational expression, ensuring that it is in its simplest form. The process involved factoring the denominators, finding the least common denominator, rewriting the fractions with the LCD, subtracting the numerators, and simplifying the resulting expression. Each step is crucial for arriving at the correct answer.
In conclusion, subtracting rational expressions requires a systematic approach. By mastering the techniques of factoring, finding the LCD, and simplifying expressions, you can confidently tackle these types of problems. This guide has provided a detailed walkthrough of the process, ensuring a clear understanding of each step involved.
