1.設f(x)為一個三次多項式,已知f(0)=f(1)=f(2)=1,f(-1)=-5,求f(x)=? 請展開 2.設多項式f(x)分別除以x平方+x-2,x平方-x-6,x平方+x-12所得餘式依次為2x+5,3x+a,4x+b,試求數對(a,b)=？ 求各位大大幫忙需詳解?

1)
設 f(x) = ax(x - 1)(x - 2) + 1
則 f(-1) = - a(- 2)(- 3) + 1 = - 5
- 6a + 1 = - 5
a = 1
f(x) = x(x - 1)(x - 2) + 1 = x(x² - 3x + 2) + 1 = x³ - 3x² + 2x + 1

2)
f(x) = Q(x) (x² + x - 2) + 2x + 5 = P(x) (x² - x - 6) + 3x + a = S(x) (x² + x - 12) + 4x + b
f(x) = Q(x) (x - 1)(x + 2) + 2x + 5 = P(x) (x - 3)(x + 2) + 3x + a = S(x) (x - 3)(x + 4) + 4x + b
f(-2) = 2(-2) + 5 = 3(-2) + a ⇒ a = 7
f(3) = 3(3) + a = 4(3) + b ⇒ b = a - 3 = 4
(a , b) = (7 , 4)