Find the roots of the equation $\displaystyle z^{10}-z^{5}-992= 0,$ whose real part is negative.
- $\displaystyle \sqrt[5]{31}\left ( \cos 108^{\circ}+i\sin 108^{\circ} \right ), -\sqrt[5]{31}, \sqrt[5]{31}\left ( \cos 252^{\circ}+i\sin 252^{\circ} \right ).$
- $\displaystyle \sqrt[5]{21}\left ( \cos 108^{\circ}+i\sin 108^{\circ} \right ), -\sqrt[5]{21}, \sqrt[5]{21}\left ( \cos 252^{\circ}+i\sin 252^{\circ} \right ).$
- $\displaystyle \sqrt[5]{31}\left ( \cos 108^{\circ}-i\sin 108^{\circ} \right ), -\sqrt[5]{31}, \sqrt[5]{31}\left ( \cos 252^{\circ}-i\sin 252^{\circ} \right ).$
- $\displaystyle \sqrt[5]{31}\left ( \cos 108^{\circ}+i\sin 108^{\circ} \right ), \sqrt[5]{31}, \sqrt[5]{31}\left ( \cos 252^{\circ}+i\sin 252^{\circ} \right ).$
Let u = z^5. u^2 - u - 992 = 0. Roots are u = (1 +/- sqrt(1 + 3968))/2 = (1 +/- 63)/2. u = 32 or u = -31. z^5 = 32 gives z = 2, 2w, 2w^2, 2w^3, 2w^4. z^5 = -31 gives z = -31^(1/5) * (cos(180+360k)/5 + i sin(180+360k)/5). The roots with negative real parts correspond to the second case.
We can factor the equation z^10 - z^5 - 992 = 0 by substituting w = z^5, resulting in the quadratic equation w^2 - w - 992 = 0. Using the quadratic formula, we find the roots for w are 32 and -31. For the positive real root w = 32, we solve z^5 = 32 using De Moivre's Theorem, yielding a modulus of the fifth root of 32 and arguments of 0, 72, 144, 216, and 288 degrees. The negative real part requires angles in the second and third quadrants, so we select 144 degrees and 216 degrees, which correspond to cos(144) + i*sin(144) and cos(216) + i*sin(216). Because the fifth root of 32 is 2 and cos(144) is equivalent to -cos(36), we can convert the angles to their principal positive equivalents to match the provided options, leading to the fifth root of 32 factored out. For the negative real root w = -31, the equation z^5 = -31 has one real root where the argument is 180 degrees, resulting in the fifth root of 31 multiplied by (cos 180 + i*sin 180), which is simply the negative fifth root of 31. Therefore, the roots with negative real parts are the fifth root of 31(cos 108 + i*sin 108), -fifth root of 31, and the fifth root of 31(cos 252 + i*sin 252).