HMMT 二月 2006 · 代数 · 第 8 题
HMMT February 2006 — Algebra — Problem 8
题目详情
- Solve for all complex numbers z such that z + 4 z + 6 = z.
解析
- Solve for all complex numbers z such that z + 4 z + 6 = z. √ √ 1 ± i 7 − 1 ± i 11 Answer: , 2 2 2 2 Solution: Rewrite the given equation as ( z + 2) + 2 = z . Observe that a solution 2 to z + 2 = z is a solution of the quartic by substitution of the left hand side into itself. √ 2 1 ± i 7 2 2 This gives z = . But now, we know that z − z + 2 divides into ( z + 2) − z + 2 = 2 4 2 2 2 4 2 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