HMMT 二月 2006 · 代数 · 第 8 题

英文解析

8. Solve for all complex numbers z such that z + 4 z + 6 = z.
   √ √
   1 ± i 7 − 1 ± i 11
   Answer: ,
   2 2
   Solution: Rewrite the given equation as ( z + 2) + 2 = z . Observe that a solution 22
   to z + 2 = z is a solution of the quartic by substitution of the left hand side into itself.2
   √
   1 ± i 72
   2 2
   This gives z = . But now, we know that z − z + 2 divides into ( z + 2) − z + 2 =
   4 2 2 2 4 22
   z + 4 z − z + 6. Factoring it out, we obtain ( z − z + 2) ( z + z + 3) = z + 4 z − z + 6.
   √
   − 1 ± i 11
   Finally, the second term leads to the solutions z = . 2