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Chapter 2: Problem 85

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

Short Answer

Expert verified
From the Rational Root Theorem, the possible zeros are -2, -1, 1 and 2. Using synthetic division or graphing, no practical rational zeros can be found. The function \(f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2\) does not have any rational zeros.

Step by step solution

01

Apply the Rational Root Theorem

The Rational Root Theorem states that if a polynomial has a rational zero \(\frac{p}{q}\), \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. Here, the constant term is 2 and the leading coefficient is 2. Therefore, the possible rational zeros of \(f(x)\) will be \(\pm 1, \pm 2\).
02

Use Synthetic Division

Apply synthetic division to test which of the possible zeros are actually zeros of the function. For each possible zero, \(c\), create a row of coefficients from the polynomial, and then proceed with synthetic division. If the result at the end is 0, then \(c\) is a zero of the polynomial. If any of them turns out to be a zero, then divide the whole polynomial by \(x-c\) and repeat synthetic division on the resulting quotient to find other zeros.
03

Consult a Graphing Utility

If the number of possible zeros is large, or if irrational or complex zeros are suspected, graph the function using the graphing calculator. Any x-intercepts on the graph represent zeros of the function. The Synthetic Division step verifies these zeros algebraically.

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Most popular questions from this chapter

The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$

Determine whether the statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(-x)$$
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