Question

If $x^2-\sqrt2 x+2=0$ has roots 𝜶 and 𝛽 then 𝜶$^{14}$ + 𝛽$^{14}$ is:

• A. -256
• B. -128
• C. $-128\sqrt2$
• D. $-256\sqrt2$

Answer 1

Answer: (B)

-128

Answer 2

The correct option is (B) : -128
$\alpha,\beta=\frac{\sqrt2+i\sqrt6}{2}$
$=\frac{\sqrt2}{2}(1+\sqrt3i)$
$=\sqrt2(\cos\frac{\pi}{3}±i\sin\frac{\pi}{3})$
$\alpha^{14}+\beta^{14}=2^7 [ (\cos\frac{14\pi}{3}+i\sin\frac{14\pi}{3})+(\cos\frac{14\pi}{3}-i\sin\frac{14\pi}{3})]$
$2^7.2\cos\frac{2\pi}{3}=-128$