Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$h(x)=\frac{x^{2}-x-20}{x^{2}+7}$$

Step-by-Step Solution

Answer

The zeros of the function \(h(x) = \frac{x^{2}-x-20}{x^{2}+7}\) are \(x = 5\) and \(x = -4\).

Step 1: Solve the numerator

Firstly, set the numerator of the function \(h(x)\) equal to zero and solve for \(x\). This can be done by factoring the quadratic equation \(x^{2}-x-20 = 0\). The factored form will be \((x - 5)(x + 4) = 0\).

Step 2: Solve for x

Now solve the equation for \(x\). Doing this, we'll get \(x = 5\) and \(x = -4\) as the roots of the equation.

Step 3: Check the denominator

We need to ensure that the denominator does not equal zero at \(x = 5\) and \(x = -4\). Substituting these values into the equation \(x^{2}+7\), it's clear that the denominator does not equal zero for any real numbers.

Step 4: Verify using a graphing utility

Finally, we can use a graphing utility to graph the function \(h(x)=\frac{x^{2}-x-20}{x^{2}+7}\). The graph should intersect with the x-axis at \(x = 5\) and \(x = -4\), which confirms that these values are indeed the zeros of the function.

Key Concepts

Rational Function Graphing

Graphing rational functions is an essential tool for understanding their characteristics. These functions, expressed as the ratio of two polynomials, can have zeros, asymptotes, and vary in end behavior. To find the zeros of a rational function, we focus on the points where the function intersects the x-axis. This occurs when the numerator is zero, and the denominator is non-zero.

For the given function, \(h(x)=\frac{x^{2}-x-20}{x^{2}+7}\), the zeros will be the solutions to the numerator equation \(x^{2}-x-20=0\) when the denominator \(x^{2}+7\) is not zero. In graphing such functions, it's helpful to also look for vertical asymptotes (set by the denominator) and horizontal asymptotes, determined by the degrees of the numerator and denominator. Remember, a rational function will never cross its vertical asymptotes, as these represent values that make the function undefined.

Factoring Quadratic Equations

Factoring quadratic equations is a fundamental skill in algebra that can simplify the process of finding the zeros of functions. A quadratic equation has the general form \(ax^2 + bx + c = 0\). Factoring involves rewriting this trinomial as the product of two binomials. This method works well when the equation is factorable over the integers.

For the equation \(x^2 - x - 20 = 0\), we search for two numbers that multiply to -20 (the constant term) and add to -1 (the coefficient of the x term). After finding the numbers -5 and +4, we can write the equation as \( (x - 5)(x + 4) = 0 \). Setting each binomial equal to zero gives us the solutions \(x = 5\) and \(x = -4\), which are the zeros of the rational function. Regular practice with different types of quadratic equations enhances the ability to quickly recognize the factors.

Graphing Utility Verification

Graphing utilities serve as a powerful verification tool when working with functions. After obtaining theoretical results analytically, a graphing utility can provide a visual confirmation of these solutions. It displays the behavior of the function across the coordinate plane and highlights important features such as intercepts, turning points, and asymptotes.

For our function \(h(x)\), after factoring and solving the quadratic equation in the numerator, we expect zeros at \(x = 5\) and \(x = -4\). By inputting the function into the graphing utility, we should see the graph crossing the x-axis at these points. Additionally, since a rational function's denominator can dictate vertical asymptotes, ensuring the denominator does not equal zero at our found zeros is crucial. In this case, the denominator is greater than zero for all real numbers, confirming the function is defined at our zeros. If a graphing utility shows matching results to our calculations, it heightens our confidence in the correctness of the solved zeros.