HMMT 十一月 2011 · 团队赛 · 第 2 题

英文解析

2. [ 4 ] Determine the set of all real numbers p for which the polynomial Q ( x ) = x + px − px − 1 hasthree distinct real roots.
   Answer: p > 1 and p < − 3 First, we note that
   3 2 2
   x + px − px − 1 = ( x − 1)( x + ( p + 1) x + 1).
   Hence, x + ( p + 1) x + 1 has two distinct roots. Consequently, the discriminant of this equation must 2
   be positive, so ( p + 1) − 4 > 0, so either p > 1 or p < − 3. However, the problem specifies that the 2
   quadratic must have distinct roots (since the original cubic has distinct roots), so to finish, we need tocheck that 1 is not a double root–we will do this by checking that 1 is not a root of x + ( p + 1) x + 12
   for any value p in our range. But this is clear, since 1 + ( p + 1) + 1 = 0 ⇒ p = − 3, which is not in theaforementioned range. Thus, our answer is all p satisfying p > 1 or p < − 3.
   4 3 2