HMMT 十一月 2011 · 团队赛 · 第 2 题

HMMT November 2011 — Team Round — Problem 2

[ 4 ] Determine the set of all real numbers p for which the polynomial Q ( x ) = x + px − px − 1 has three distinct real roots. 4 3 2

解析

[ 4 ] Determine the set of all real numbers p for which the polynomial Q ( x ) = x + px − px − 1 has three 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). 2 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 quadratic must have distinct roots (since the original cubic has distinct roots), so to finish, we need to 2 check that 1 is not a double root–we will do this by checking that 1 is not a root of x + ( p + 1) x + 1 for any value p in our range. But this is clear, since 1 + ( p + 1) + 1 = 0 ⇒ p = − 3, which is not in the aforementioned range. Thus, our answer is all p satisfying p > 1 or p < − 3. 4 3 2