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Chapter 2: Problem 46

Use synthetic division to divide. $$\frac{3 x^{3}-4 x^{2}+5}{x-\frac{3}{2}}$$

Short Answer

Expert verified
After the synthetic division, the polynomial \(3x^3 - 4x^2 + 5\) divided by \(x - \frac{3}{2}\) results in \(3x^2 -1x+ \frac{1}{2}\) with remainder 5.

Step by step solution

01

Prepare the synthetic division.

First, write down the coefficients of the polynomial. From the given \(3x^3 - 4x^2 + 5\), the coefficients are 3, -4, 0, and 5 (0 is for the missing term for \(x\)). Put these coefficients in a row. Put \(a = \frac{3}{2}\) to the left and draw a line below them.
02

Begin the synthetic division.

Start by bringing down the first coefficient (which is 3) below the line. Then, multiply this by \(a\), and write the result under the second coefficient, then add them and write the sum under the line. Repeat this process for each pair of numbers (sum and next coefficient): Multiply the sum by \(a\), then add the resulting product and the next coefficient, and write the new sum below the line.
03

Write the result.

The numbers obtained below the line are the coefficients of the quotient. If the original polynomial has degree n, then the quotient will have degree \(n-1\). The last number is the remainder. In this case, the polynomial being divided is of degree 3, so the quotient will be of degree 2. Thus, the result of the division is a quadratic equation plus the remainder divided by the divisor.

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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.

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) Zero \(-3+\sqrt{2} i\)

Find the rational zeros of the polynomial function. $$f(z)=z^{3}+\frac{11}{6} z^{2}-\frac{1}{2} z-\frac{1}{3}=\frac{1}{6}\left(6 z^{3}+11 z^{2}-3 z-2\right)$$

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-2 x^{3}-3 x^{2}+12 x-18\) (Hint: One factor is \(\left.x^{2}-6 .\right)\)
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