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Chapter 2: Problem 61

Use synthetic division to show that \(x\) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. $$2 x^{3}-15 x^{2}+27 x-10=0, \quad x=\frac{1}{2}$$

Short Answer

Expert verified
After performing synthetic division and confirming \(x=\frac{1}{2}\) as a root, factoring the polynomial completely will give the reduced polynomial \(2x^{2} - 14x + 20\). Using the quadratic formula, the two remaining roots are found to be \(x=4\) and \(x=2.5\). So, the real solutions of the equation are \(x=\frac{1}{2}\), \(x=2.5\), and \(x=4\).

Step by step solution

01

Perform Synthetic Division

Start by setting up the synthetic division with \(x=\frac{1}{2}\) as the divisor and \(2, -15, 27, -10\) as the coefficients of the polynomial. Synthetic division should end with a remainder of zero if \(x=\frac{1}{2}\) is indeed a root.
02

Confirm if \(x=\frac{1}{2}\) is a Root

After performing synthetic division, confirm if \(x=\frac{1}{2}\) is a root of the polynomial equation. This is true if the remainder of the synthetic division is zero. If so, then \(x=\frac{1}{2}\) is a root of the polynomial equation.
03

Factor the Polynomial Completely

Utilize the result of the synthetic division to factor the polynomial completely. The quotient from the synthetic division describes the factors of the polynomial.
04

Find the Remaining Roots

The factored equation gained from the synthetic division result will have a quadratic factor which could be factored further or the roots can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). By doing this, the remaining real roots of the polynomial equation will be found.

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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.

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

Think About It \(\quad\) Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive and that has a zero at \(x=3\) of multiplicity 2

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=2 x^{3}-3 x^{2}-3$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{2}+10 x+17$$
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