In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form.
x³ − (1 + i√3) = 0

Verified step by step guidance

1. Rewrite the given equation as \(x^{3} = 1 + i\sqrt{3}\) to isolate \(x^{3}\) on one side.
2. Express the complex number on the right side, \(1 + i\sqrt{3}\), in polar form by finding its magnitude \(r\) and argument \(\theta\). Calculate the magnitude as \(r = \sqrt{1^{2} + (\sqrt{3})^{2}}\) and the argument as \(\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right)\).
3. Use De Moivre's Theorem to find the cube roots of the complex number. The general formula for the \(n\)th roots is \(x_k = r^{1/3} \left( \cos\left(\frac{\theta + 2k\pi}{3}\right) + i \sin\left(\frac{\theta + 2k\pi}{3}\right) \right)\) for \(k = 0, 1, 2\).
4. Calculate each root \(x_k\) in polar form by substituting \(k = 0, 1, 2\) into the formula to find the three distinct solutions.
5. Convert each polar form solution \(x_k = r^{1/3} (\cos \phi + i \sin \phi)\) into rectangular form using \(x_k = r^{1/3} \cos \phi + i r^{1/3} \sin \phi\) by evaluating the cosine and sine values.

Key Concepts

Complex Numbers in Rectangular and Polar Form

Complex numbers can be expressed in rectangular form as a + bi, where a is the real part and b is the imaginary part. Alternatively, they can be represented in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). Converting between these forms is essential for solving and expressing solutions clearly.

Roots of Complex Numbers and De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form, raising it to a power n involves raising the magnitude to n and multiplying the angle by n. Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, producing multiple solutions spaced evenly around the circle.

Solving Polynomial Equations in the Complex Plane

Polynomial equations with complex coefficients can have multiple complex roots. To solve equations like x³ = c, express c in polar form, then find all cube roots by applying the nth root formula. Each root corresponds to a distinct solution in both rectangular and polar forms.