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Chapter 2: Problem 97

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{4}-4 x^{3}+16 x-16\) (a) Upper: \(x=5\) (b) Lower: \(x=-3\)

Short Answer

Expert verified
By using synthetic division, we verified that \(x=5\) is indeed an upper bound and \(x=-3\) is a lower bound for the real zeros of the given polynomial function.

Step by step solution

01

Understanding synthetic division

Synthetic division is a shortcut method to divide a polynomial by a linear polynomial of the form \(x-a\). It simplifies the process by only using the coefficients of the polynomials.
02

Synthetic division for upper bound (x = 5)

First, we arrange the polynomial in descending order with \[f(x)=x^{4}-4 x^{3}+0x^{2}+16 x-16\]. Always remember to include zero in your synthetic division if necessary. Then, let's set up the synthetic division with '5' at the box and carry out the operations: \[\begin{{array}}{{r|rrrrr}}5 & 1 & -4 & 0 & 16 & -16 \ & & 5 & 5 & 25 & 125 \\hline & 1 & 1 & 5 & 41 & 109 \\end{{array}}\]All signs are positive, so x=5 is an upper bound.
03

Synthetic division for lower bound (x = -3)

Setting up the synthetic division with '-3' at the box and carry out the operations:\[\begin{{array}}{{r|rrrrr}}-3 & 1 & -4 & 0 & 16 & -16 \ & & -3 & 21 & -63 & 141 \\hline & 1 & -7 & 21 & -47 & 125 \\end{{array}}\]The signs are alternating, indicating x=-3 is a lower bound.

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Most popular questions from this chapter

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. $$f(x)=x^{3}+24 x^{2}+214 x+740$$

Simplify the complex number and write it in standard form. $$(-i)^{6}$$

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-2 x$$

The numbers \(N\) (in millions) of students enrolled in schools in the United States from 2000 through 2009 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number, \(N\) } \\\\\hline 2000 & 72.2 \\\2001 & 73.1 \\\2002 & 74.0 \\\2003 & 74.9 \\\2004 & 75.5 \\\2005 & 75.8 \\\2006 & 75.2 \\\2007 & 76.0 \\\2008 & 76.3 \\\2009 & 77.3 \\\\\hline\end{array}$$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=0\) corresponding to 2000. (b) Use the regression feature of the graphing utility to find a quartic model for the data. (A quartic model has the form \(a t^{4}+b t^{3}+c t^{2}+d t+e,\) where \(a, b\) \(c, d, \text { and } e \text { are constant and } t \text { is variable. })\) (c) Graph the model and the scatter plot in the same viewing window. How well does the model fit the data? (d) According to the model, when did the number of students enrolled in schools exceed 74 million? (e) Is the model valid for long-term predictions of student enrollment? 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. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$
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