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Chapter 9: Problem 15

Show that the polynomial \(z^{3}-z^{2}+4 z+5\) has all its roots in the disk \(|z|<3\)

Short Answer

Expert verified
By applying Rouche's theorem, all roots of the polynomial \(z^3 - z^2 + 4z + 5\) are within the disk \(|z|<3\).

Step by step solution

01

Identify the Polynomial

The given polynomial is \(p(z) = z^3 - z^2 + 4z + 5\). To apply Rouche鈥檚 Theorem, we need to express this as \(p(z) = f(z) + g(z)\).
02

Express the Polynomial in the form \(f(z) + g(z)\) to Apply Rouche鈥檚 Theorem

Let's define \(f(z) = z^3 + 4z\) and \(g(z) = -z^2 + 5\). So the polynomial can be written as \(p(z) = f(z) + g(z)\).
03

Apply Rouche's Theorem

Define a circle \(C\): \(|z| = 3\). Now, we need to demonstrate that for each point on \(C\), \(|f(z)|>|g(z)|\). On \(C\), we find that \(|f(z)| = |z|^3 + 4|z| = 3^3 + 4*3 = 27 + 12 = 39\) and \(|g(z)| = |-z|^2 + 5 = 9+5 = 14\). Clearly, \(|f(z)| > |g(z)|\). Therefore, by Rouche's theorem, \(p(z) = f(z) + g(z)\) and \(f(z)\) have the same number of roots inside the disk \(|z| < 3\). Since \(f(z) = z^3 + 4z\) obviously has three roots inside this disk (0 and the roots of z^2 = -4), this means that \(p(z)\) must also have three roots in the disk.

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Key Concepts

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

Complex Polynomial Roots
Understanding complex polynomial roots is a fundamental aspect of higher mathematics. When we speak of roots in the context of complex polynomials, we are referring to solutions of the equation \( p(z) = 0 \), where \( p(z) \) is a polynomial with complex coefficients. These roots may be real, complex, or a mixture of both.

For a polynomial of degree \( n \), there will always be \( n \) roots when counted with their multiplicities, a principle known as the Fundamental Theorem of Algebra. Locating these roots can often be challenging, which is where tools like Rouche's theorem come into play.
Rouche's Theorem Application
Rouche's Theorem is a powerful tool in complex analysis for determining the number of zeros of a complex function within a particular region. In order to apply Rouche's Theorem, we write the given polynomial \( p(z) \) as \( p(z) = f(z) + g(z) \) where both \( f(z) \) and \( g(z) \) are complex functions.

Conditions for Application

  • Both functions \( f(z) \) and \( g(z) \) must be analytic on a disk and its boundary.
  • On the boundary of the disk, the inequality \( |f(z)| > |g(z)| \) must hold for all points.
If these conditions are met, then \( f(z) \) and \( p(z) \) have the same number of zeros inside the disk. This result significantly simplifies the task of locating the polynomial's roots.
Polynomial Factorization
Polynomial factorization is the process of breaking down a complex polynomial into a product of simpler polynomials, ideally linear and/or quadratic factors, which are easier to solve. Factors are polynomials of lesser degree that multiply to give the original polynomial. The roots of the original polynomial correspond to the roots of its factors.

In practice, factorizing simple polynomials directly is often possible, but for higher degree polynomials or those with complex coefficients, methods like synthetic division or using theorems like Rouche's can be necessary to simplify the problem.
Complex Analysis
Complex analysis is the study of functions that take complex numbers as inputs and provide complex numbers as outputs. Being a highly rich field, it delves into complex functions, their properties, and the analytical techniques useful in exploring these functions.

Part of its toolkit includes the study of mappings, contour integrals, and theorems like Rouche's which allow mathematicians and engineers to explore complex dynamics and solve problems involving complex polynomials, such as finding their roots within certain regions, as seen in the exercise we used Rouche's Theorem on.

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Most popular questions from this chapter

If \(f(z)\) has poles at a sequence of points \(\left\\{z_{n}\right\\}\), and \(z_{n} \rightarrow z_{0}\), show that \(f(z)\) does not have a pole at \(z=z_{0} .\) Illustrate this fact by a concrete example.

Show that the polynomial \(z^{4}+4 z-1\) has one root in the disk \(|z|<1 / 3\) and the remaining three roots in the annulus \(1 / 3<|z|<2\).

Evaluate the following integrals along different simple closed curves not passing through 0 and \(\pm 1\). (i) \(\int_{C} \frac{e^{z}-1}{z^{2}(z-1)} d z\) (ii) \(\int_{C} \frac{e^{z}}{z^{2}\left(1-z^{2}\right)} d z\).

Using the argument principle, prove the Fundamental Theorem of Algebra.

Given arbitrary distinct complex numbers \(z_{0}, z_{1}\) and \(z_{2}\), construct a function \(f(z)\) having a removable singularity at \(z=z_{0}\), a pole of order \(k\) at \(z=z_{1}\), and an essential singularity at \(z=z_{2}\).
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