Question

One of the roots of the equation \((x + 1)^4 + 81 = 0\) is

• A. \(3\left(\frac{1+i}{\sqrt{2}}\right)\)
• B. \(-\left(\frac{3+\sqrt{2}+3i}{\sqrt{2}}\right)\)
• C. \(-\left(\frac{3+\sqrt{2}+i}{\sqrt{2}}\right)\)
• D. \(-\left(\frac{3+3i}{\sqrt{2}}\right)\)

Hint:

For equations of the form \(z^n = -a\), the roots will always lie symmetrically on a circle of radius \(a^{1/n}\) in the complex plane. Checking angles corresponding to \(k = 0, 1, 2, \dots\) sequentially is the most reliable way to match with options.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the roots of the given complex polynomial equation. We can rearrange the equation and use De Moivre's theorem to find all four roots.
Step 2: Key Formula or Approach:
Express the right side in polar form and find the roots using the fractional power: \[ z = r(\cos \theta + i \sin \theta) \implies z^{1/n} = r^{1/n} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right] \] for \(k = 0, 1, 2, \dots, n-1\).
Step 3: Detailed Explanation:
Rearranging the equation: \[ (x + 1)^4 = -81 \] We can write \(-81\) in polar form as \(81(\cos \pi + i \sin \pi)\).
Taking the fourth root on both sides: \[ x + 1 = 81^{1/4} \left[ \cos\left(\frac{\pi + 2k\pi}{4}\right) + i\sin\left(\frac{\pi + 2k\pi}{4}\right) \right] \] where \(k = 0, 1, 2, 3\).
Since \(81^{1/4} = 3\), the roots for \(x+1\) are: \[ x+1 = 3 \left( \cos\frac{(2k+1)\pi}{4} + i \sin\frac{(2k+1)\pi}{4} \right) \] Let's evaluate for \(k = 2\): \[ x+1 = 3 \left( \cos\frac{5\pi}{4} + i \sin\frac{5\pi}{4} \right) \] Since \(\cos\frac{5\pi}{4} = -\frac{1}{\sqrt{2}}\) and \(\sin\frac{5\pi}{4} = -\frac{1}{\sqrt{2}}\): \[ x+1 = 3 \left( -\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \right) = \frac{-3 - 3i}{\sqrt{2}} \] Solving for \(x\): \[ x = \frac{-3 - 3i}{\sqrt{2}} - 1 = \frac{-3 - 3i - \sqrt{2}}{\sqrt{2}} = -\left(\frac{3 + \sqrt{2} + 3i}{\sqrt{2}}\right) \] This matches Option (B).
Step 4: Final Answer:
One of the roots of the given equation is \(-\left(\frac{3+\sqrt{2}+3i}{\sqrt{2}}\right)\).