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Chapter 2: Problem 38

Use synthetic division to divide. $$\frac{x^{5}-13 x^{4}-120 x+80}{x+3}$$

Short Answer

Expert verified
The result of the synthetic division is \(-16x^{4} + 48x^{3} - 144x^{2} + 432x - 1296 + \frac{144}{x+3}\).

Step by step solution

01

Setup the synthetic division table

Write down the coefficients of the dividend, \(x^{5}-13x^{4}-120x+80\), which are \(1, -13, 0, 0, -120, 80\), to the right of vertical bar. On the left of the bar, write down the opposite of the value of x for which the divisor, \(x+3\), equals 0, which is \(-3\).
02

Perform synthetic division

Bring down the first coefficient (\(1\)) to the bottom row. Multiply this number by \(-3\) (from left of the bar) to obtain \(-3\), write this below the second coefficient (\(-13\)). Add these to obtain \(-16\), write this in the bottom row. Repeat this: Multiply \(-16\) by \(-3\) to get \(48\), add this to \(0\) to obtain \(48\). Continue the process until you've performed calculations for all coefficients.
03

Write the final answer

The last number in the bottom row (\(144\)) is the remainder. The other numbers from left to right in the bottom row are the coefficients of the result \(-16, 48, -144, 432, -1296\). Start the polynomial with the highest power (\(x^{4}\), because the degree of the original polynomial is \(5\) and we divided by \(x+3\), which has degree \(1\)), such that the coefficients match to the numbers in the bottom row. The result ends up to be \(-16x^{4} + 48x^{3} - 144x^{2} + 432x - 1296 + \frac{144}{x+3}\).

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Most popular questions from this chapter

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. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)

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

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.

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.

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