Problem 115 - Precalculus with Limits

Question

THINK ABOUT IT Explain how you can use DeMoivre's Theorem to solve the polynomial equation \(x^4 + 16 = 0\). [Hint: Write \(-16\) as \(16(\cos\ \pi + i\ \sin\ \pi)\).]

Step-by-Step Solution

Answer

The roots for the equation \(x^4 + 16 = 0\) are \(2\sqrt{2}\ (\cos\ \frac{\pi}{4} + i\ \sin\ \frac{\pi}{4})\), \(2\sqrt{2}\ (\cos\ \frac{3\pi}{4} + i\ \sin\ \frac{3\pi}{4})\), \(2\sqrt{2}\ (\cos\ \frac{5\pi}{4} + i\ \sin\ \frac{5\pi}{4})\), and \(2\sqrt{2}\ (\cos\ \frac{7\pi}{4} + i\ \sin\ \frac{7\pi}{4})\).

Step 1: Rewrite the Equation

First, rewrite the given equation as \(x^4 = -16\). Now, use the hint and write -16 in the form of a trigonometric complex number. So, \(x^4 = 16(\cos\ \pi + i\ \sin\ \pi)\). This might look tricky, but remember that cosine of \(\pi\) is -1 and sine of \(\pi\) is 0.

Step 2: Apply DeMoivre's Theorem

Now, we can apply DeMoivre's Theorem. We have \(x = \sqrt[4]{16} \cdot \left( \cos \left( \frac{ \pi + 2\pi k}{4} \right) + i \sin \left( \frac{ \pi + 2\pi k}{4} \right) \right)\), where k = 0, 1, 2, 3. Here, \(k\) represents the four different fourth roots of the complex number.

Step 3: Calculate the roots

Substitute the different values of \(k\) into the equation. Calculating the sine and cosine values for each root, we get four solutions: x = \(2\sqrt{2}\ (\cos\ \frac{\pi}{4} + i\ \sin\ \frac{\pi}{4})\), x = \(2\sqrt{2}\ (\cos\ \frac{3\pi}{4} + i\ \sin\ \frac{3\pi}{4})\), x = \(2\sqrt{2}\ (\cos\ \frac{5\pi}{4} + i\ \sin\ \frac{5\pi}{4})\), and x = \(2\sqrt{2}\ (\cos\ \frac{7\pi}{4} + i\ \sin\ \frac{7\pi}{4})\).

Key Concepts

Complex Numbers in Trigonometric Form

Understanding complex numbers in their trigonometric form is pivotal to solving equations where roots are not readily apparent. Imagine complex numbers as points in a plane, with a real axis (horizontal) and an imaginary axis (vertical). Trigonometric form introduces a new way to represent these numbers that ties into their geometric interpretation.

In trigonometric form, a complex number is expressed as \( r(\text{cos} \theta + i\sin \theta) \), where \( r \) is the magnitude of the complex number and \( \theta \) is the angle it forms with the positive real axis—also known as the argument of the complex number. When a negative number like \(-16\) is represented in trigonometric form, we can see it as a vector with a magnitude of 16 that points directly left along the real axis, giving it an angle of \( \pi \).

This representation becomes particularly useful when dealing with polynomial equations that have complex solutions. It enables the application of DeMoivre's Theorem, which provides a methodical approach to finding the roots of a complex number by linking them with the properties of sine and cosine. This connects algebraic operations to geometric implications, such as rotations and rescaling, enhancing our understanding of how complex numbers behave.

Polynomial Equation Solving

Solving polynomial equations, especially those resulting in complex solutions, can be challenging. Traditional methods taught in algebra provide sufficient tools for simple cases, but when it comes to higher-degree polynomials or those leading to complex roots, we must look for more advanced techniques. One such approach involves leveraging the trigonometric form of complex numbers and applying DeMoivre's Theorem.

When faced with an equation like \(x^4 + 16 = 0\), the process begins by isolating the term with the exponent and finding its roots. By doing so, we often encounter complex numbers when the term is negative, as even roots of negative numbers are not real. At this step, rewriting the negative term in trigonometric form—as the problem suggests for \(-16\)—sets the stage for applying DeMoivre's theorem. This theorem allows us to find all the roots of the equation by dividing the argument of the complex number by the degree of the polynomial. Through this process, we can systemically calculate each root, complex or real, for any polynomial equation.

Roots of Complex Numbers

The roots of complex numbers can seem like a mysterious topic, but with DeMoivre's theorem, they are unraveled with elegance and precision. DeMoivre's theorem states that for a given complex number \( z = r(\text{cos} \theta + i\sin \theta) \) and a positive integer \( n \), the \( n^{th} \) roots are given by \( \sqrt[n]{r} \text{cos} \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \), where \( k \) is an integer from \( 0 \) to \( n - 1 \).

In practice, for our equation \( x^4 + 16 = 0 \), this means dividing the angle \( \pi \) by 4, which is the degree of the polynomial, and finding four different roots corresponding to four equally spaced points on the unit circle. This geometric perspective not only aids in calculation but also enriches our intuition on the symmetry and properties of complex numbers. It is this intersection of algebraic understanding and geometric visualization that makes working with complex numbers, and discovering their roots, an engrossing journey through different fields of mathematics.