Simplify The Rational Expression (5x - X^2) / (x^2 - 8x + 15) And Identify Undefined X-Values

In this comprehensive guide, we will delve into the process of simplifying the rational expression ${\frac{5x - x^2}{x^2 - 8x + 15}}$ and, more importantly, pinpoint the x-values that render the expression undefined. Mastering these techniques is crucial for success in algebra and calculus, providing a solid foundation for more advanced mathematical concepts. We'll break down each step, ensuring a clear understanding of the underlying principles. Our goal is to provide you with a thorough understanding of how to simplify rational expressions and identify values that make them undefined. This involves factoring, canceling common factors, and understanding the significance of the denominator in a rational expression.

Understanding Rational Expressions

Before we begin, it's essential to grasp what rational expressions are. A rational expression is essentially a fraction where the numerator and denominator are polynomials. In our case, both ${5x - x^2}$ and ${x^2 - 8x + 15}$ are polynomials. The key characteristic of rational expressions is that they can be simplified, much like regular fractions. However, we must always be mindful of the denominator. Division by zero is undefined in mathematics, so any value of x that makes the denominator zero will make the entire expression undefined. This is why identifying these values is a critical part of the simplification process. We will explore this in detail, showing you how to systematically find these critical values and understand their implications. By the end of this guide, you will not only be able to simplify this specific expression but also apply the same techniques to a wide variety of rational expressions.

Step-by-Step Simplification

Let's embark on the journey of simplifying ${\frac{5x - x^2}{x^2 - 8x + 15}}$. The first step is to factor both the numerator and the denominator. Factoring is the process of breaking down a polynomial into its constituent factors, which are expressions that, when multiplied together, give the original polynomial. For the numerator, ${5x - x^2}$, we can factor out an x, which gives us ${x(5 - x)}$. Factoring is a critical skill in algebra, and it's essential for simplifying rational expressions. It allows us to identify common factors that can be canceled out, making the expression simpler. This step not only simplifies the expression but also reveals potential values of x that would make the denominator zero, which are the values we need to avoid. We will discuss the importance of identifying these values and how they relate to the domain of the rational expression.

Factoring the Numerator

Taking a closer look at the numerator, ${5x - x^2}$, we identify the greatest common factor (GCF) as x. Factoring out x, we get ${x(5 - x)}$. This simple step is the foundation for further simplification. Notice that we now have two factors in the numerator: x and ${(5 - x)}$. Factoring is like reverse multiplication; it helps us break down complex expressions into simpler components. The factored form of the numerator makes it easier to see if there are any common factors with the denominator, which is our next step. This technique is widely used in algebra to simplify expressions and solve equations. Understanding how to factor polynomials is crucial for success in higher-level mathematics.

Factoring the Denominator

Now, let's turn our attention to the denominator, ${x^2 - 8x + 15}$. This is a quadratic expression, and we need to find two numbers that multiply to 15 and add up to -8. These numbers are -3 and -5. Therefore, we can factor the denominator as ${(x - 3)(x - 5)}$. Factoring quadratic expressions is a fundamental skill in algebra, and there are various methods to do it, including the trial-and-error method, the quadratic formula, and completing the square. In this case, we used the trial-and-error method, which involves finding the factors of the constant term (15) and testing different combinations until we find a pair that adds up to the coefficient of the linear term (-8). The factored form of the denominator is crucial because it directly reveals the values of x that would make the denominator zero, which are the values we need to exclude from the domain of the rational expression.

Rewriting the Expression

After factoring both the numerator and the denominator, our expression now looks like this:

$$\frac{x(5 - x)}{(x - 3)(x - 5)}$$

This rewritten form is much easier to work with because it reveals the underlying structure of the expression. We can now clearly see the factors in both the numerator and the denominator, which allows us to identify common factors that can be canceled out. This is the core idea behind simplifying rational expressions. By factoring and canceling common factors, we can reduce the expression to its simplest form, making it easier to analyze and work with. The process of rewriting the expression in its factored form is a crucial step in simplifying rational expressions and identifying undefined values.

Identifying and Canceling Common Factors

At this stage, we observe a potential for simplification. Notice that we have ${(5 - x)}$ in the numerator and ${(x - 5)}$ in the denominator. These factors are very similar but not exactly the same. However, we can manipulate one of them to match the other. We can factor out a -1 from ${(5 - x)}$, which gives us ${-1(x - 5)}$. This is a common trick used in simplifying rational expressions, and it's important to understand why it works. By factoring out a -1, we change the order of the terms inside the parentheses, which allows us to match the factor in the denominator. Now, our expression looks like this:

$$\frac{x(-1)(x - 5)}{(x - 3)(x - 5)}$$

Now we have a clear common factor of ${(x - 5)}$ in both the numerator and the denominator. We can cancel out this common factor, which simplifies the expression further. Canceling common factors is a fundamental step in simplifying rational expressions, and it's based on the principle that dividing a quantity by itself results in 1. However, it's crucial to remember that we can only cancel factors, not terms. This means that we can cancel expressions that are multiplied together, but not expressions that are added or subtracted. The process of identifying and canceling common factors is a key step in simplifying rational expressions and identifying undefined values.

Simplified Expression

After canceling the common factor ${(x - 5)}$, we are left with:

$$\frac{-x}{x - 3}$$

This is the simplified form of the original expression. Simplification is a critical process in mathematics because it makes expressions easier to understand and work with. A simplified expression is equivalent to the original expression, but it is in a more concise and manageable form. This is particularly important in calculus and other advanced mathematical fields, where complex expressions need to be simplified before they can be manipulated. The simplified expression ${\frac{-x}{x - 3}}$ is much easier to analyze and work with than the original expression, especially when we need to find the values of x that make the expression undefined.

Identifying Undefined Values

Now, the crucial step: identifying the x-values for which the expression is undefined. A rational expression is undefined when its denominator equals zero. This is because division by zero is undefined in mathematics. Therefore, we need to find the values of x that make the denominator of the original expression, ${x^2 - 8x + 15}$, equal to zero. It's important to use the original denominator, not the simplified one, because canceling factors can sometimes hide the values that make the expression undefined. In this case, the original denominator is ${(x - 3)(x - 5)}$, so we need to find the values of x that make this product equal to zero.

Setting the Denominator to Zero

To find the undefined values, we set the original denominator equal to zero:

$$x^2 - 8x + 15 = 0$$

We already factored this quadratic expression as ${(x - 3)(x - 5) = 0}$. Now, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero:

$$x - 3 = 0 \text{ or } x - 5 = 0$$

Solving these equations gives us the values of x that make the denominator zero.

Solving for x

Solving the equation ${x - 3 = 0}$, we add 3 to both sides, which gives us ${x = 3}$. This means that when x is 3, the denominator of the original expression becomes zero, making the expression undefined. Similarly, solving the equation ${x - 5 = 0}$, we add 5 to both sides, which gives us ${x = 5}$. This means that when x is 5, the denominator of the original expression also becomes zero, making the expression undefined. These values, x = 3 and x = 5, are critical points for the rational expression, and they are excluded from the domain of the function.

Undefined Values

Therefore, the expression ${\frac{5x - x^2}{x^2 - 8x + 15}}$ is undefined when x = 3 and x = 5. These values make the denominator zero, resulting in an undefined expression. It's crucial to identify these values because they represent points where the function is not defined. In the context of graphing rational functions, these values correspond to vertical asymptotes, which are vertical lines that the graph of the function approaches but never touches. Understanding undefined values is essential for accurately analyzing and working with rational expressions and functions. This knowledge is particularly important in calculus, where we often need to find the limits of functions and identify discontinuities.

Conclusion

In conclusion, we have successfully simplified the rational expression ${\frac{5x - x^2}{x^2 - 8x + 15}}$ to ${\frac{-x}{x - 3}}$ and identified the x-values for which the expression is undefined, which are x = 3 and x = 5. This process involved factoring the numerator and denominator, canceling common factors, and setting the original denominator equal to zero to find the undefined values. These are fundamental techniques in algebra and calculus that are essential for working with rational expressions and functions. By mastering these techniques, you will be well-equipped to tackle more complex mathematical problems and gain a deeper understanding of the behavior of rational functions. The ability to simplify rational expressions and identify undefined values is a valuable skill that will serve you well in your mathematical journey.

By following this step-by-step guide, you've gained a solid understanding of how to simplify rational expressions and identify values that make them undefined. These skills are not just important for algebra; they form the foundation for more advanced topics in mathematics and other quantitative fields.