Simplifying The Difference Of Rational Expressions

In the realm of mathematics, particularly in algebra, rational expressions play a crucial role. These expressions, which are essentially fractions with polynomials in the numerator and denominator, can often appear complex and daunting. However, by understanding the fundamental principles of algebraic manipulation, we can unravel their intricacies and simplify them effectively. This article will delve into the process of simplifying and finding the difference between two rational expressions, providing a step-by-step guide and illuminating the underlying concepts. Our journey will involve factoring, finding common denominators, and performing algebraic operations, all essential tools in the mathematician's arsenal.

When dealing with rational expressions, the first step often involves simplifying each expression individually. This typically means factoring the polynomials in the numerator and denominator to identify any common factors that can be canceled out. Factoring is the process of breaking down a polynomial into its constituent factors, which are expressions that, when multiplied together, yield the original polynomial. There are various techniques for factoring, including factoring out the greatest common factor (GCF), using the difference of squares formula, and employing the quadratic formula for quadratic expressions. Once the expressions are simplified, we can then proceed to find a common denominator, which is a crucial step in adding or subtracting rational expressions. The common denominator is the least common multiple (LCM) of the denominators of the expressions being combined. By expressing each rational expression with the common denominator, we can then perform the addition or subtraction operation on the numerators, while keeping the common denominator unchanged. This process allows us to combine the expressions into a single, simplified rational expression.

Algebraic manipulation is the key to success when working with rational expressions. It requires a solid understanding of factoring techniques, the ability to identify common denominators, and the skill to perform addition, subtraction, multiplication, and division of polynomials. By mastering these fundamental skills, we can confidently tackle complex rational expressions and simplify them to their most basic forms. This not only makes the expressions easier to work with but also provides deeper insights into their underlying mathematical structure. Moreover, the ability to simplify rational expressions is essential in various areas of mathematics, including calculus, differential equations, and abstract algebra. It is a foundational skill that empowers us to solve a wide range of mathematical problems and to appreciate the beauty and elegance of algebraic manipulations.

Deconstructing the Problem

Let's consider the problem at hand: finding the difference between the rational expressions $\frac{x}{x^2-2 x-15}$ and $\frac{4}{x^2+2 x-35}$. This problem exemplifies the challenges and rewards of working with rational expressions. To solve it, we must systematically apply our knowledge of factoring, finding common denominators, and algebraic manipulation. The first step, as always, is to factor the denominators of the expressions. This will allow us to identify the factors that are common to both denominators and the factors that are unique to each. By understanding the factored form of the denominators, we can then determine the least common multiple (LCM), which will serve as our common denominator.

Factoring the denominators involves breaking them down into their simplest factors. The denominator $x^2 - 2x - 15$ can be factored as $(x - 5)(x + 3)$, while the denominator $x^2 + 2x - 35$ can be factored as $(x + 7)(x - 5)$. Notice that both denominators share a common factor of $(x - 5)$. This is a crucial observation, as it simplifies the process of finding the common denominator. The least common multiple (LCM) of the two denominators is the product of all the unique factors, each raised to the highest power that appears in any of the denominators. In this case, the LCM is $(x - 5)(x + 3)(x + 7)$. This expression will serve as our common denominator, allowing us to combine the two rational expressions.

Once we have the common denominator, we must rewrite each rational expression with this denominator. This involves multiplying the numerator and denominator of each expression by the factors that are missing from its original denominator. For the first expression, $\frac{x}{(x - 5)(x + 3)}$, we need to multiply the numerator and denominator by $(x + 7)$. For the second expression, $\frac{4}{(x + 7)(x - 5)}$, we need to multiply the numerator and denominator by $(x + 3)$. This process ensures that both expressions have the same denominator, allowing us to subtract them. After rewriting the expressions with the common denominator, we can then proceed to subtract the numerators, simplifying the resulting expression as much as possible. This may involve combining like terms, factoring the numerator, and canceling out any common factors between the numerator and denominator. The final result will be the simplified difference between the two original rational expressions.

Step-by-Step Solution

Now, let's walk through the solution step-by-step. First, we identify the two rational expressions: $\frac{x}{x^2-2 x-15}$ and $\frac{4}{x^2+2 x-35}$. As discussed earlier, the first step is to factor the denominators. We have already established that $x^2 - 2x - 15$ factors as $(x - 5)(x + 3)$ and $x^2 + 2x - 35$ factors as $(x + 7)(x - 5)$. This gives us the expressions $\frac{x}{(x - 5)(x + 3)}$ and $\frac{4}{(x + 7)(x - 5)}$.

The next step is to find the common denominator. As we determined earlier, the least common multiple (LCM) of the denominators is $(x - 5)(x + 3)(x + 7)$. Now we rewrite each rational expression with this common denominator. For the first expression, we multiply the numerator and denominator by $(x + 7)$:

$\frac{x}{(x - 5)(x + 3)} \cdot \frac{(x + 7)}{(x + 7)} = \frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)}$

For the second expression, we multiply the numerator and denominator by $(x + 3)$:

$\frac{4}{(x + 7)(x - 5)} \cdot \frac{(x + 3)}{(x + 3)} = \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}$

Now that both expressions have the same denominator, we can subtract them:

$\frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)} - \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}$

This simplifies to:

$\frac{x(x + 7) - 4(x + 3)}{(x - 5)(x + 3)(x + 7)}$

Expanding the numerator gives us:

$\frac{x^2 + 7x - 4x - 12}{(x - 5)(x + 3)(x + 7)}$

Combining like terms in the numerator yields:

$\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$

This is our simplified expression. At this point, we should check if the numerator can be factored further. In this case, the quadratic $x^2 + 3x - 12$ does not factor easily using integer coefficients. Therefore, we can leave the expression as it is. The final simplified difference between the two rational expressions is:

$\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$

Comparing the Results

Now, let's compare our result with the expressions provided in the original problem. We have arrived at the expression $\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$. The problem also presents several other expressions:

• $\frac{x^2+3 x+12}{(x-3)(x-5)(x+7)}$
• $\frac{x(x+3-12)}{(x+3)(x-5)(x+7)}$
• $\frac{x^2+3 x+12}{(x+3)(x-5)(x+7)}$
• $\frac{x^2+3 x-12}{(x+3)(x-5)(x+7)}$

Notice that our final answer, $\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}$, closely matches the last expression in the list, $\frac{x^2+3 x-12}{(x+3)(x-5)(x+7)}$. The only difference is the order of the factors in the denominator, which does not affect the value of the expression since multiplication is commutative. This confirms that our solution is correct.

The other expressions in the list represent intermediate steps or potential errors in the simplification process. For example, the expression $\frac{x^2+3 x+12}{(x-3)(x-5)(x+7)}$ has a different quadratic in the numerator, indicating a possible mistake in expanding or combining terms. The expression $\frac{x(x+3-12)}{(x+3)(x-5)(x+7)}$ seems to be an attempt to factor or simplify the expression, but it is not equivalent to the correct answer. By comparing these expressions, we can appreciate the importance of careful and methodical algebraic manipulation.

Conclusion

In conclusion, finding the difference between rational expressions involves a series of steps, including factoring the denominators, finding a common denominator, rewriting the expressions with the common denominator, subtracting the numerators, and simplifying the result. By carefully applying these steps, we can successfully navigate the complexities of rational expressions and arrive at the correct solution. The problem we have explored, finding the difference between $\frac{x}{x^2-2 x-15}$ and $\frac{4}{x^2+2 x-35}$, serves as a valuable example of the techniques and strategies involved in working with rational expressions. Through this process, we have not only solved a specific problem but also deepened our understanding of algebraic manipulation and the fundamental principles of mathematics.