哪位老师在,求所有实系数多项式P(t),使所有实数t都满足tP(t-1)=(t-2)P(t)

Question

哪位老师在,求所有实系数多项式P(t),使所有实数t都满足tP(t-1)=(t-2)P(t)

Answer 1

实系数多项式P(t),对所有实数t都满足tP(t-1)=(t-2)P(t) ,①
t=0时P(0)=0,
t=1时P(1)=0,
∴设P(t)=t(t-1)*Q(t),
代入①,t*(t-1)(t-2)*Q(t-1)=(t-2)*t(t-1)*Q(t),
∴Q(t-1)=Q(t)为常数c,
∴P(t)=ct(t-1).