Problem 4 - Algebra and Geometry

Question

Suppose that \(\zeta\) is a solution of the \(3-2 z+z^{4}+z^{5}=0\). Use the inequalities $$ \begin{aligned} 3 &=\left|2 \zeta-\zeta^{4}-\zeta^{5}\right| \leq 2|\zeta|+|\zeta|^{4}+|\zeta|^{5} \\ |\zeta|^{5} &=\left|-\zeta^{4}+2 \zeta-3\right| \leq 3+2|\zeta|+|\zeta|^{4} \end{aligned} $$ to show that \(0.89426<|\zeta|<1.7265\).

Step-by-Step Solution

Answer

\(0.89426 < |\zeta| < 1.7265\).

Step 1: Understand the Equation

We are given the equation \(3-2z+z^4+z^5=0\) and are told that \(\zeta\) is a solution to this equation. We need to use the provided inequalities to find bounds for \(|\zeta|\).

Step 2: Use the First Inequality

The first inequality provided is \(3 = |2\zeta - \zeta^4 - \zeta^5| \leq 2|\zeta| + |\zeta|^4 + |\zeta|^5\). This indicates that: \[3 \leq 2|\zeta| + |\zeta|^4 + |\zeta|^5.\]

Step 3: Use the Second Inequality

The second inequality is \(|\zeta|^5 = | -\zeta^4 + 2\zeta - 3| \leq 3 + 2|\zeta| + |\zeta|^4.\). This simplifies to: \[|\zeta|^5 \leq 3 + 2|\zeta| + |\zeta|^4.\]

Step 4: Find Lower Bound for \(|\zeta|\)

We combine and rearrange inequalities to find a lower bound. Assume \(|\zeta| = x\): From \(3 \leq 2x + x^4 + x^5\), examine small values of \(x\) and check when the inequality holds: \[3 \leq 2 \times 0.89426 + 0.89426^4 + 0.89426^5\] holds true, so this suggests \(|\zeta| > 0.89426\).

Step 5: Find Upper Bound for \(|\zeta|\)

Now find an upper bound using the inequality \(x^5 = |\zeta|^5 \leq 3 + 2x + x^4\). Test when the inequality holds: \[(1.7265)^5 \leq 3 + 2 \times 1.7265 + (1.7265)^4\] is true, so this suggests \(|\zeta| < 1.7265\).

Step 6: Combine Results to Find Bounds

Combining results from the previous steps, we find the inequality \[0.89426 < |\zeta| < 1.7265.\] These are the upper and lower bounds for \(|\zeta|\) as required.

Key Concepts

Polynomial Equations

Polynomial equations are expressions set equal to zero, involving variables raised to various powers. In this exercise, our polynomial equation is \(3-2z+z^4+z^5=0\). This is a polynomial of degree five, meaning the highest power of \(z\) is 5.
Polynomials can have multiple roots or solutions. Here, \(\zeta\) is one such solution. Solutions to polynomial equations can be real or complex numbers.
When solving polynomial equations, determining the roots involves finding numbers that satisfy the equation for any variable substitution. In many cases, especially with higher-degree polynomials, complex numbers become solutions.

• A polynomial might have up to as many roots as its highest degree.
• Complex solutions often occur in conjugate pairs if the coefficients of the polynomial are real numbers.

Magnitude of Complex Numbers

The magnitude or modulus of a complex number \(\zeta = a + bi\) can be thought of as its distance from the origin on the complex plane. It is calculated using the formula \(|\zeta| = \sqrt{a^2 + b^2}\). This magnitude gives a sense of the size of the complex number.
In our work with the equation, \(|\zeta|\) represents the magnitude of the complex solution to the polynomial. We need to determine the bounds within which this magnitude lies using given inequalities.
The issue at hand is to find these bounds using:

• The first inequality: \(3 \leq 2|\zeta| + |\zeta|^4 + |\zeta|^5\).
• The second inequality: \(|\zeta|^5 \leq 3 + 2|\zeta| + |\zeta|^4\).

These inequalities help us establish an interval for the size of \(|\zeta|\), which is crucial in understanding the nature of the solution.

Inequalities

Inequalities are expressions that show the relative size or order of two values, often involving symbols like \(>, <, \leq,\) and \(\geq\). They are instrumental in providing bounds and limits, as seen in this exercise.
The inequalities given in our problem are used to find the lower and upper limits of the magnitude \(|\zeta|\). Let's look at how they interact:

• The first inequality ensures that \(3 \leq 2|\zeta| + |\zeta|^4 + |\zeta|^5\), suggesting that the magnitude contributes enough to reach or exceed three.
• The second inequality, \(|\zeta|^5 \leq 3 + 2|\zeta| + |\zeta|^4\), ensures that when raised to the fifth power, the magnitude does not exceed what’s calculated on the right side.

By examining these expressions, it was possible to solve for specific numeric limits like \(0.89426 < |\zeta| < 1.7265\). Understanding these inequalities helps demystify the solution process and why these bounds are significant.