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Chapter 3: Problem 3
Determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x$$
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The functions $$f(x)=0.0875 x^{2}-0.4 x+66.6$$ and $$g(x)=0.0875 x^{2}+1.9 x+11.6$$ model a car's stopping distance, \(f(x)\) or \(g(x),\) in feet, traveling at \(x\) miles per hour. Function \(f\) models stopping distance on dry pavement and function g models stopping distance on wet pavement. The graphs of these functions are shown for \(\\{x | x \geq 30\\} .\) Notice that the figure does not specify which graph is the model for dry roads and which is the model for wet roads. Use this information to solve. (GRAPH CANNOT COPY). a. Use the given functions to find the stopping distance on dry pavement and the stopping distance on wet pavement for a car traveling at 55 miles per hour. Round to the nearest foot. b. Based on your answers to part (a), which rectangular coordinate graph shows stopping distances on dry pavement and which shows stopping distances on wet pavement? c. How well do your answers to part (a) model the actual stopping distances shown in Figure 3.43 on page \(411 ?\) d. Determine speeds on wet pavement requiring stopping distances that exceed the length of one and one-half football fields, or 540 feet. Round to the nearest mile per hour. How is this shown on the appropriate graph of the models?
Determine whether cach statement is true or false If bhe statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.
In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ and let \(\frac{P}{q}\) be a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as $$a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}$$ b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{P}{q}\) is reduced to lowest terms, \(p\) and \(q\) have no common factors other than \(-1\) and 1 Because \(p\) does divide the right side and has no factors in common with \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n}\).
Use the four-step procedure for solving variation problems given on page 424 to solve. The distance that a spring will stretch varies directly as the force applied to the spring. A force of 12 pounds is needed to stretch a spring 9 inches. What force is required to stretch the spring 15 inches?
What is a rational function?
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