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Chapter 3: Problem 2
Determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=7 x^{2}+9 x^{4}$$
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Will help you prepare for the material covered in the next section. a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)
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?
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 Exercises 1-10. \(y\) varies directly as \(x . y=65\) when \(x=5 .\) Find \(y\) when \(x=12 .\)
Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies jointly as \(z\) and the difference between \(y\) and \(w\).
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