HKALE2006 純粹數學試題求解

已知p(x)為四次多項式,p(0)=0 p(1)=1/2 p(2)=2/3 p(3)=3/4 p(4)=4/5
q(x)=(x+1)p(x)-x

(a)i. 求q(0),q(1),q(2),q(3), q(4)
ii. 以x次數為1的多項式的積表示q(x)
q(0)=p(0)=0
q(1)=2p(1)-1=0
q(2)=3p(2)-2=0
q(3)=0
q(4)=0
(ii)
since 1,2,3,4 are the roots of q(x) and the degree of q(x) is 5
q(x)=x(x-1)(x-2)(x-3)(x-4)

2006-12-07 00:54:38 補充：
好明顯上面個位ii做錯

2006-12-07 00:55:32 補充：
寫漏了since 0,1,2,3,4.........

You are also incorrect. You forgot the leading coefficient of q(x).　
Let q(x)=Ax(x-1)(x-2)(x-3)(x-4)=(x+1)p(x)-x　
Put x=-1,A(-1)(-2)(-3)(-4)(-5)=1　
A=-1/120

(a)
q(0)=(0+1)p(0)-0=1*0-0=0
q(1)=(1+1)p(1)-1=2*1/2-1=0
q(2)=(2+1)p(2)-2=3*2/3-2=0
q(3)=(3+1)p(3)-3=4*3/4-3=0
q(4)=(4+1)p(4)-4=5*4/5-4=0
(b)
p(x)=x/(x+1)
q(x)=(x+1)*x/(x+1)-x=x-x=0