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Chapter 3: Problem 94

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{3}-14 x+8$$

Short Answer

Expert verified
The complex zeros are \(x = 2, \; -1 + \sqrt{5}, \; -1 - \sqrt{5} \).

Step by step solution

01

Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial \(f(x) = x^3 - 14x + 8\) must be a factor of the constant term (8) divided by a factor of the leading coefficient (1). The potential rational roots are \(\text{卤1, 卤2, 卤4, 卤8} \).
02

Test possible rational roots

Test the possible rational roots using synthetic division. First, let's test \(x = 1\). Perform the synthetic division: \[ \begin{array}{r|rrr} 1 & 1 & 0 & -14 & 8 \ & & 1 & 1 & -13 & -5 \ \hline & 1 & 1 & -13 & -5 \end{array} \] Since the remainder is not zero, \(x = 1 \) is not a root. Repeat the process for \(x = -1, 2, -2, 4, -4, 8, -8 \).
03

Confirm root found

Testing \(x = 2\): \[ \begin{array}{r|rrr} 2 & 1 & 0 & -14 & 8 \ & & 2 & 4 & -20 & -24 \ \hline & 1 & 2 & -10 & 0 \end{array} \] The remainder is zero, so \(x = 2\) is a root.
04

Factor out the root

Since \(x = 2\) is a root, we can factor out \(x - 2\): \ \(f(x) = (x - 2)(x^2 + 2x - 4)\).
05

Solve the quadratic equation

Solve the quadratic equation \(x^2 + 2x - 4 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1, b = 2, c = - 4\). \ \ \[x = \frac{-2 \pm \sqrt{4 + 16}}{2} = \frac{-2 \pm \sqrt{20}}{2} = \frac{-2 \pm 2\sqrt{5}}{2} = -1 \pm \sqrt{5} \] Thus, the roots are \(x = -1 \pm \sqrt{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Root Theorem
The Rational Root Theorem is a valuable tool in determining potential rational zeros of a polynomial. Here鈥檚 how it works: If you have a polynomial equation like \( f(x) = x^3 - 14x + 8 \), the theorem tells us that any rational root, \( p/q \), must be a factor of the constant term (here 8) divided by a factor of the leading coefficient (here 1).
So, for our example, the possible rational roots are:
  • \( \pm 1 \)
  • \( \pm 2 \)
  • \( \pm 4 \)
  • \( \pm 8 \)
By testing these values, we can find out if they are indeed the roots of the polynomial.
Synthetic Division
Once you have potential rational roots, synthetic division helps quickly verify if they are actual roots. Synthetic division is an efficient way to divide polynomials. Let鈥檚 break it down through an example. Suppose we want to test whether \( x = 2 \) is a root of \( f(x) = x^3 - 14x + 8 \). We set up the synthetic division:
  • Write down the coefficients of the polynomial: \( 1, 0, -14, 8 \)
  • Place 2 (the potential root) to the left
  • Perform the synthetic division process, where we bring down the first coefficient (1), and then repeatedly multiply and add:
\ \begin{array}{r|rrr} 2 & 1 & 0 & -14 & 8 \ & & 2 & 4 & -20 & -24 \ \hline & 1 & 2 & -10 & 0 \end{array}
The remainder is 0, confirming that \( x = 2 \) is indeed a root. When the remainder is zero, it shows that the value you tested is a root. If it鈥檚 not zero, you just move on to the next potential root.
Quadratic Formula
After finding a root using the Rational Root Theorem and synthetic division, factor it out. You might be left with a quadratic equation. In our example, factoring out \( x - 2 \) gives us \( (x - 2)(x^2 + 2x - 4) \). To find the remaining roots, solve the quadratic equation using the quadratic formula:
  • The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • Identify the coefficients: for \( x^2 + 2x - 4 \) , \( a = 1, b = 2, \) and \( c = -4 \)
Plug these into the formula:
\[ x = \frac{-2 \pm \sqrt{4 + 16}}{2} \]
Simplify further:
\[ x = \frac{-2 \pm \sqrt{20}}{2} = \frac{-2 \pm 2 \sqrt{5}}{2} = -1 \pm \sqrt{5} \]
Thus, the quadratic formula gives us the remaining complex zeros \( x = -1 \pm \sqrt{5} \). Using these steps, you can systematically find complex zeros for any polynomial.

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Most popular questions from this chapter

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. A. The \(x\) -intercept is \(-3\) B. The \(y\) -intercept is 5 C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a "hole" in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote, and the \(y\) -axis is not its vertical asymptote. H. The \(x\) -axis is its horizontal asymptote, and the \(y\) -axis is its vertical asymptote. $$f(x)=\frac{x+7}{x+1}$$

Solve each problem. AIDS Deaths in the United States The table* lists the total (cumulative) number of known deaths caused by AIDS in the United States up to 2007 (a) Plot the data. Let \(x=0\) correspond to the year 1990 . (b) Would a linear or a quadratic function model the data better? Explain. (c) Find a quadratic function defined by \(g(x)=a x^{2}+b x+c\) that models the data. (d) Plot the data together with \(g\) on the same coordinate plane. How well does \(g\) model the number of AIDS cases? (e) Use \(g\) to estimate the total number of AIDS deaths in the year 2010 . (f) Consider the last two entries in the table for the years 2006 and 2007 . Is it safe to assume that the quadratic model given for \(g(x)\) will continue for years 2008 and beyond? $$\begin{array}{c|c||c|c} \hline \text { Year } & \text { AIDS Deaths } & \text { Year } & \text { AIDS Deaths } \\ \hline 1990 & 119,821 & 1999 & 419,234 \\ \hline 1991 & 154,567 & 2000 & 436,373 \\ \hline 1992 & 191,508 & 2001 & 454,099 \\ \hline 1993 & 220,592 & 2002 & 471,417 \\ \hline 1994 & 269,992 & 2003 & 489,437 \\ \hline 1995 & 320,692 & 2004 & 507,536 \\ \hline 1996 & 359,892 & 2005 & 524,547 \\ \hline 1997 & 381,738 & 2006 & 565,927 \\ \hline 1998 & 400,743 & 2007 & 583,298 \\ \hline \end{array}$$

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=12 x^{4}-43 x^{3}+50 x^{2}+38 x-12$$

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. What is the \(y\) -intercept of the graph of \(f ?\)

Work each problem.What happens to \(y\) if \(y\) varies inversely as \(x,\) and \(x\) is doubled?
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