Problem
4) \( x^{6}+65 x^{3}+64=0(4) \) A) \( \left\{-4,-2+i \sqrt{11},-2-i \sqrt{11},-1, \frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}\right\} \) B) \( \left\{-4, \frac{-1+i \sqrt{47}}{3}, \frac{-1-i \sqrt{47}}{3},-1, \frac{1+i \sqrt{11}}{2}, \frac{1-i \sqrt{1}}{2}\right. \) C) \( \{0,-4,2+2 i \sqrt{3}, 2-2 i \sqrt{3},-1,1\} \) D) \( \left\{-4,2+2 i \sqrt{3}, 2-2 i \sqrt{3},-1, \frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}\right\} \) \( -1- \)
Solution
Step 1 ：Let \(u = x^3\), then \(u^2 + 65u + 64 = 0\)
Step 2 ：Factor the quadratic: \((u + 1)(u + 64) = 0\)
Step 3 ：Solve for \(u\) and then for \(x\): \(u = -1, -64\) which yields \(x = \left\{-4, 2+2i\sqrt{3}, 2-2i\sqrt{3}, -1, \frac{1+i\sqrt{3}}{2}, \frac{1-i\sqrt{3}}{2}\right\}\)
