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Chapter 3: Problem 108

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=4 x^{4}-65 x^{2}+16$$

Short Answer

Expert verified
The complex zeros are \[ x = 4, -4, \frac{1}{2}, -\frac{1}{2} \].

Step by step solution

01

- Identify the Form of the Polynomial

Observe that the polynomial is given by \(f(x)=4x^{4}-65x^{2}+16\). Notice that the polynomial is in quadratic form with respect to \(x^2\).
02

- Substitute to Simplify the Polynomial

To make solving easier, let \(u = x^2\), so the polynomial becomes \(4u^2 - 65u + 16\).
03

- Solve the Quadratic Equation

Use the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 4\), \(b = -65\), and \(c = 16\). Plug in these values to get: \[ u= \frac{65 \pm \sqrt{(-65)^2 - 4 \cdot 4 \cdot 16}}{2 \cdot 4} = \frac{65 \pm \sqrt{4225 - 256}}{8} = \frac{65 \pm \sqrt{3969}}{8} = \frac{65 \pm 63}{8} \] This simplifies to two solutions: \(u = \frac{128}{8} = 16\) and \(u = \frac{2}{8} = \frac{1}{4}\).
04

- Substitute Back for x and Solve

Recall that \(u = x^2\). Therefore, we solve for \(x\) in \(x^2 = 16\) and \(x^2 = \frac{1}{4}\): \[ x^2 = 16 \Rightarrow x = \pm 4 \] \(x^2 = \frac{1}{4} \Rightarrow x = \pm \frac{1}{2}\).
05

- Write the Final Answer

Collect all the solutions: \[ x = 4, -4, \frac{1}{2}, -\frac{1}{2} \]

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Key Concepts

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

quadratic form
When dealing with polynomial equations, a quadratic form is one where we can rewrite the equation in the format of a quadratic equation. In our example, the original polynomial given was: \[ f(x) = 4x^4 - 65x^2 + 16 \] To notice the quadratic form, we observe that the powers of x are multiples of 2. Here, we rewrite the equation in terms of \(x^2\), making it easier to solve using quadratic methods. Think of it like this: substituting \(u = x^2\) transforms the polynomial into a simpler equation: \[ 4u^2 - 65u + 16 \] Recognizing quadratic forms allows us to apply familiar techniques, such as the quadratic formula, which simplifies the problem-solving process significantly.
substitution method
The substitution method is a vital tool for simplifying complex polynomial equations. By substituting \(u = x^2\), we reduce the complexity of the polynomial. Here's a step-by-step breakdown:
  • First, we identify a simpler variable to replace a more complex expression. In our case, we let \(u = x^2\).
  • This transforms our polynomial from \(4x^4 - 65x^2 + 16\) to \(4u^2 - 65u + 16\).
  • Now, solving the quadratic equation in \(u\) form becomes straightforward.
Finally, after solving for \(u\), we revert our substitution by replacing \(u\) back with \(x^2\). This step enables us to find the actual roots of the original polynomial.
quadratic formula
The quadratic formula is a powerful tool specifically designed to solve quadratic equations of the form \(au^2 + bu + c = 0\). The formula is: \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our exercise, the quadratic equation derived after substitution was \(4u^2 - 65u + 16\). We identified the coefficients as:
  • a = 4
  • b = -65
  • c = 16
Substituting these values into the quadratic formula, we solve for \(u\): \[ u = \frac{65 \pm \sqrt{4225 - 256}}{8} = \frac{65 \pm \sqrt{3969}}{8} = \frac{65 \pm 63}{8} \] Simplifying, we derive two solutions: \(u = 16\) and \(u = \frac{1}{4}\). The quadratic formula is invaluable for finding precise roots of polynomial equations.
solving polynomial equations
Solving polynomial equations involves finding the values of the variable that satisfy the equation. In our case, here鈥檚 the comprehensive process:
  • First, identify the polynomial and determine if it can be transformed (or simplified). Our example was in quadratic form, so we substituted \(u = x^2\), reducing it to \(4u^2 - 65u + 16\).
  • Next, use the quadratic formula to find solutions for \(u\). We found \(u = 16\) and \(u = \frac{1}{4}\).
  • Then, revert the substitution by solving \(x^2 = 16\), giving \(x = \pm 4\). For \(x^2 = \frac{1}{4}\), the solutions are \(x = \pm \frac{1}{2}\).
  • Combine all roots for the final answer: \(x = 4, -4, \frac{1}{2}, -\frac{1}{2}\).
Following these steps ensures a methodical approach to solving polynomial equations.

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Most popular questions from this chapter

In \(1545,\) a method of solving a cubic equation of the form $$x^{3}+m x=n$$ developed by Niccolo Tartaglia, was published in the Ars Magna, a work by Girolamo Cardano. The formula for finding the one real solution of the equation is $$x=\sqrt[3]{\frac{n}{2}+\sqrt{\left(\frac{n}{2}\right)^{2}+\left(\frac{m}{3}\right)^{3}}}-\sqrt[3]{\frac{-n}{2}+\sqrt{\left(\frac{n}{2}\right)^{2}+\left(\frac{m}{3}\right)^{3}}}$$ (Source: Gullberg, J., Mathematics from the Birth of Numbers, W.W. Norton \& Company.) Use the formula to solve each equation for the one real solution. $$x^{3}+9 x=26$$

Solve each problem.Current in a Circuit Speed of a Pulley The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

Explain how the graph of each function can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Then graph \(f\) and give the (a) domain and (b) range. Determine the intervals of the domain for which the function is ( \(c\) ) increasing or (d) decreasing. See Examples \(1-3\). $$f(x)=\frac{1}{(x-3)^{2}}$$

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. $$f(x)=\frac{x^{2}-16}{x+4}$$

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 are the \(x\) -intercepts of the graph of \(f ?\)
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