100 Roots In... How Many Roots Out? #jeeadvanced #pymt

Welcome to PYMT #26 (Problems You Must Try). Today's problem comes from *JEE Advanced Study Material* and begins with a seemingly innocent statement: A polynomial of degree 100 has 100 distinct real roots. But then comes the real challenge: How many real roots does the equation 100P(x)·P''(x) = 99(P'(x))² have? Question: Let $P(x)$ is polynomial of degree 100 and the equation \(P(x) =0 \quad \text{has 100 real and distinct roots.} \) Find the number of real roots of the equation \[ 100 P(x) \cdot P''(x) = 99 (P'(x))^2. \] At first glance, the problem appears computational and intimidating. Yet hidden inside it is a beautiful interaction between algebra, calculus, and the structure of polynomial roots. What makes this problem special is not just the answer, but the journey. The solution reveals a remarkably elegant pattern that transforms a difficult-looking equation into something surprisingly manageable. If you enjoy mathematical thinking, problem solving, and discovering beautiful ideas hidden beneath technical expressions, this problem is for you. 📌 Source: JEE Advanced Study Material If you enjoyed the video, consider subscribing and sharing it with someone who loves mathematics. #PYMT #JEEAdvanced #Polynomial #Calculus #Algebra #ProblemSolving #Mathematics #MathOlympiad #StayMathactive