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Chapter 2: Problem 80

(a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine the exact value of one of the zeros, and (c) use synthetic division to verify your result from part (b), and then factor the polynomial completely. $$g(x)=6 x^{4}-11 x^{3}-51 x^{2}+99 x-27$$

Short Answer

Expert verified
The roots of the polynomial \(g(x)=6x^4 -11x^3 -51x^2 +99x -27\) can be obtained via graphing and then, one root can be exactly determined by using the Rational Root Theorem. Synthetic Division is further used to confirm the validity of the root and fully factorize the polynomial.

Step by step solution

01

Approximate the roots by graphing

Draw the graph of the function \(g(x)=6x^4 -11x^3 -51x^2 +99x -27\). Utilize your graphing tool's zero or root feature to estimate the roots. Assume, the root values to be \(r_1\), \(r_2\), \(r_3\) and \(r_4\). Every root value should be precise up to three decimal points.
02

Calculate one root algebraically

The root value can be calculated algebraically. Use the Rational Root Theorem, which states that any rational root, written in lowest terms, has a numerator that is a factor of the constant term, and a denominator that is a factor of the leading coefficient. By applying this theorem, try a few numbers and plug those into the equation until the equation equals zero. Let's say we have one of the roots as \(r\).
03

Use synthetic division to verify the root

Use the root found in Step 2 to perform synthetic division on the polynomial. The remainder should be zero. This confirms that the roots found by the first two methods are indeed roots of the original polynomial.
04

Factorize the Polynomial

After confirming that the root is valid, factor out the polynomial \(g(x)\) using synthetic division's output as a guide. The factorization of the polynomial will lead to having a product of a binomial term and a cubic polynomial, which would have been impossible to factorize without synthetic division.

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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. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

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. $$f(s)=2 s^{3}-5 s^{2}+12 s-5$$

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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.
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