Finding Values Where Rational Expression (x+8)/(x^2-2x-24) Is Undefined

In mathematics, a rational expression is a fraction where the numerator and denominator are polynomials. Understanding when these expressions are undefined is crucial for various mathematical operations and problem-solving scenarios. A rational expression becomes undefined when its denominator equals zero, as division by zero is not permitted in mathematics. This article delves into the process of identifying the values of the variable that make a given rational expression undefined. We will explore the algebraic techniques used to find these values and provide a step-by-step guide to solve such problems. The specific expression we will examine is (x+8)/(x^2-2x-24), and our goal is to determine the values of x for which this expression is undefined.

To determine when a rational expression is undefined, the primary focus is on the denominator. A rational expression, fundamentally a fraction, is undefined when the denominator is equal to zero. This is because division by zero is an undefined operation in mathematics. Consider the rational expression (x+8)/(x^2-2x-24). The numerator, x + 8, does not affect whether the expression is defined or undefined. However, the denominator, x2 - 2x - 24, plays a crucial role. The expression is undefined for any value of x that makes x2 - 2x - 24 equal to zero. Therefore, to find these values, we need to solve the equation x2 - 2x - 24 = 0. This involves finding the roots of the quadratic equation, which can be done through factoring, completing the square, or using the quadratic formula. In this particular case, factoring is a straightforward method to find the solutions. By setting the denominator equal to zero and solving for x, we can identify the values that make the rational expression undefined, ensuring we avoid these values in any further calculations or analysis involving the expression.

To find the values of x for which the rational expression (x+8)/(x^2-2x-24) is undefined, we must set the denominator equal to zero and solve for x. The denominator is the quadratic expression x2 - 2x - 24. This leads us to the equation x2 - 2x - 24 = 0. The next step is to factor the quadratic expression. We are looking for two numbers that multiply to -24 and add up to -2. These numbers are -6 and 4. Thus, we can factor the quadratic expression as (x - 6)(x + 4) = 0. Now, we apply 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. This gives us two separate equations: x - 6 = 0 and x + 4 = 0. Solving the first equation, x - 6 = 0, we add 6 to both sides to get x = 6. Solving the second equation, x + 4 = 0, we subtract 4 from both sides to get x = -4. Therefore, the values of x that make the denominator zero are x = 6 and x = -4. These are the values for which the rational expression is undefined. Any other value of x will result in a non-zero denominator, and the expression will be defined.

Factoring the quadratic expression x2 - 2x - 24 is a crucial step in determining the values of x for which the rational expression is undefined. The goal of factoring is to rewrite the quadratic expression as a product of two binomials. To factor x2 - 2x - 24, we need to find two numbers that multiply to the constant term (-24) and add up to the coefficient of the x term (-2). We can start by listing pairs of factors of -24: (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), and (-4, 6). Among these pairs, the pair (4, -6) satisfies both conditions: 4 multiplied by -6 equals -24, and 4 plus -6 equals -2. Therefore, we can rewrite the middle term, -2x, as 4x - 6x. This gives us the expression x2 + 4x - 6x - 24. Now, we can factor by grouping. We group the first two terms and the last two terms: (x2 + 4x) + (-6x - 24). From the first group, we can factor out an x, giving us x(x + 4). From the second group, we can factor out a -6, giving us -6(x + 4). Now we have x(x + 4) - 6(x + 4). We can see that (x + 4) is a common factor, so we factor it out, resulting in (x - 6)(x + 4). Thus, the factored form of x2 - 2x - 24 is (x - 6)(x + 4). This factorization allows us to easily find the values of x that make the expression equal to zero by applying the zero-product property.

The zero-product property is a fundamental principle in algebra that states if the product of two or more factors is zero, then at least one of the factors must be zero. This property is particularly useful when solving equations that are in factored form, such as the one we obtained after factoring the denominator of the rational expression. In our case, the factored form of the denominator x2 - 2x - 24 is (x - 6)(x + 4). We set this equal to zero to find the values of x that make the rational expression undefined: (x - 6)(x + 4) = 0. Applying the zero-product property, we set each factor equal to zero: x - 6 = 0 and x + 4 = 0. Solving the equation x - 6 = 0, we add 6 to both sides, which gives us x = 6. This means that when x is 6, the factor (x - 6) becomes zero, and thus the entire product (x - 6)(x + 4) is zero. Similarly, solving the equation x + 4 = 0, we subtract 4 from both sides, which gives us x = -4. This means that when x is -4, the factor (x + 4) becomes zero, and the entire product is zero. Therefore, the zero-product property allows us to quickly identify the values of x that make the denominator zero, which are x = 6 and x = -4. These are the values for which the rational expression (x+8)/(x^2-2x-24) is undefined.

After factoring the denominator and applying the zero-product property, we have determined that the values of x that make the rational expression (x+8)/(x^2-2x-24) undefined are x = 6 and x = -4. These are the values that, when substituted into the denominator x2 - 2x - 24, result in zero. To verify this, we can substitute these values back into the denominator. First, let's substitute x = 6: (6)2 - 2(6) - 24 = 36 - 12 - 24 = 0. This confirms that the denominator is zero when x = 6. Next, let's substitute x = -4: (-4)2 - 2(-4) - 24 = 16 + 8 - 24 = 0. This also confirms that the denominator is zero when x = -4. Since the denominator is zero at these values, the rational expression is undefined at x = 6 and x = -4. These values must be excluded from the domain of the rational expression. The domain represents all possible values of x for which the expression is defined. In this case, the domain is all real numbers except 6 and -4. Graphically, these values would correspond to vertical asymptotes on the graph of the rational function, indicating that the function approaches infinity (or negative infinity) as x approaches these values.

Now that we have determined the values of x for which the rational expression (x+8)/(x^2-2x-24) is undefined, we can check the given options to see which ones match our solutions. The options provided are: A. 8, B. -6, C. 6, D. -8, E. -4, F. 4. We found that the expression is undefined when x = 6 and x = -4. Comparing these values with the options, we can see that options C and E match our solutions. Option C, 6, is one of the values we found, and option E, -4, is the other value. The other options (A, B, D, and F) do not match our solutions, meaning that substituting these values into the denominator would not result in zero. Therefore, the correct answers are C. 6 and E. -4. These are the only values from the given options that make the rational expression undefined.

In conclusion, to determine the values of x for which the rational expression (x+8)/(x^2-2x-24) is undefined, we focused on finding the values that make the denominator equal to zero. We set the denominator, x2 - 2x - 24, equal to zero and factored it into (x - 6)(x + 4) = 0. By applying the zero-product property, we found that the solutions are x = 6 and x = -4. These values make the denominator zero, thus rendering the rational expression undefined. When checking the given options, we identified that options C. 6 and E. -4 are the correct answers. Understanding how to identify values that make rational expressions undefined is crucial in algebra and calculus, as it helps in determining the domain of functions, simplifying expressions, and solving equations involving rational functions. This process involves factoring, applying the zero-product property, and verifying the solutions, which are fundamental skills in mathematical analysis.