Problem 17 - Contemporary Precalculus

Question

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$\frac{2 x^{5}-10 x+5}{x^{3}+x^{2}-12 x}=0 ; \quad(1, \infty)$$

Step-by-Step Solution

Answer

Answer: x = 1.26

Step 1: Rewrite the equation

To rewrite the given equation, set it as y = f(x): $$y = \frac{2 x^{5}-10 x+5}{x^{3}+x^{2}-12 x}$$

Step 2: Find the critical points

Calculate the critical points by finding when the denominator is zero: $$x^{3}+x^{2}-12 x = 0$$ Factor it, and you will find the critical points: $$x(x+4)(x-3) = 0$$ The critical points are x = 0, x = -4, and x = 3.

Step 3: Identify the open interval

The given open interval is (1, ∞). In this interval, there are no critical points of the function.

Step 4: Find the x-intercepts in the given open interval using graphical approximation

Use a graphing calculator or online tool to graph the function: $$y = \frac{2 x^{5}-10 x+5}{x^{3}+x^{2}-12 x}$$ Examine the graph in the given open interval (1, ∞) and find the approximate x-intercepts. These x-intercepts represent the solutions to the given equation in the open interval.

Step 5: State the approximate solution

After analyzing the graph and observing the x-intercepts in the open interval (1, ∞), you will find that there is one x-intercept within the interval, located approximately around x = 1.26 (rounded to two decimal places). Therefore, the approximate solution of the given equation in the open interval (1, ∞) is x = 1.26.

Key Concepts

Finding Roots of Equations

Finding the roots of an equation is all about determining the values of \(x\) that make the expression equal to zero. In many cases, a formula or algebraic approach is used to find these values. However, when faced with complex expressions, graphical methods come in handy. Using a graph allows you to visualize where the function crosses the x-axis, pointing to the roots.


When graphing, you plot the function and scan for points or intervals where the curve hits the x-axis (the horizontal line). These intersections are your roots or solutions to the equation. In situations where the expression is complex, as in rational functions, graphical approximation tools, like graph calculators or software, are often necessary to pinpoint these points accurately.

Sometimes a precise root may not be easily found, so an approximate value can offer a practical solution. This approximation still qualifies as a root where the function's value approaches zero, which is also valid as a solution in certain contexts.

Open Intervals in Calculus

An open interval is a part of calculus that involves a particular range of numbers without including the endpoints. Imagine it as the interval \((a, b)\), where neither \(a\) nor \(b\) is part of the set. In simpler terms, it's the range of values between, but not including, \(a\) and \(b\).

In our example, the open interval is \((1, \infty)\), which refers to the set of numbers greater than 1 without any upper bound—excluding 1 itself. When looking for roots within an open interval, it’s vital to focus only on the values within this specified range.


Critical points, where the denominator of a rational function may be zero, help in determining important boundaries. Yet with open intervals, any critical points outside the interval concern only the generalized function and should not affect the search within the given range. This makes understanding and clearly identifying the open interval crucial in locating roots accurately and precisely.

Graphing Rational Functions

Graphing rational functions involves plotting an expression where one polynomial is divided by another. Analyzing these functions taps into various calculus concepts like asymptotes, intercepts, and critical points.

Start by identifying any vertical asymptotes; these occur where the denominator is zero, hinting at where the function's value could become infinite. Mark these on your graph. In the equation provided, solving \(x^3 + x^2 - 12x = 0\) reveals x = 0, x = -4, and x = 3 as points to watch, though they may not all impact the given interval of interest due to its boundaries.

Next, find horizontal asymptotes, relevant when the degrees of the numerator and denominator lead the graph to settle towards constant values: they guide understanding of the graph's overall shape. Intercepts are another key feature. They're found by setting the numerator equal to zero.

Given the expression, substituting values in a graphing calculator or tool within the desired interval, such as the outlined (1, \(\infty\)), gives a visual means to locate the roots. The root, or "zero," seen at approximately \(x = 1.26\), is where the graph cuts the x-axis. This point is the practical solution within the stipulations of the initial problem.