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1.設f(x)為多項式且degf(x)>3,若f(x)之常數項為2,
且f(x)除以(x-2)(x+1)的餘式為2x-6,則f(x)除以
x(x-2)(x+1)的餘式為
2.已知多項式f(x)除以2x=-3x-2的餘式為3x-2,多項
式g(x)除以²-x-2的餘式為-3x+5 則f(x)+g(x)除以x-2
的餘式為____
3.已知四次多項式 f(x)除以x2+x+1的商式為q1(x)
餘式為3x-2,除以x-1的商式為q2(x)、餘式為7
,試回答下列問題:
(1)試問 f(x)除以x=-1的餘式?
(2)若f(x)除以x-1的商式為q3(x)
,
則q3(2)-q1(2)=

Answer 1

✨ 最佳解答 ✨

qn

1. f(x) = x(x-2)(x+1)q(x) + a(x-2)(x+1) + 2x-6
f(0) = a(-2)(1) - 6
2 = -2a-6
a=-4
餘式=-4(x-2)(x+1)+2x-6
=-4(x²-x-2)+2x-6
=-4x²+6x+2

2.
f(x) = (2x²-3x-2)q₁(x)+3x-2
g(x) = (x²-x-2)q₂(x)-3x+5
→
f(x) = (2x+1)(x-2)q₁(x)+3x-2
g(x) = (x+1)(x-2)q₂(x)-3x+5

f(x)+g(x) = (x-2)[(2x+1)q₁(x)+(x+1)q₂(x)]+3x-2-3x+5
=(x-2)[(2x+1)q₁(x)+(x+1)q₂(x)]+3
餘式即為3

3.(1)
f(x) = (x-1)(x²+x+1)(ax+b) + c(x²+x+1) + 3x-2
f(1) = c(1+1+1) + 3-2
7=3c+1
c=2
→ f(x)=(x-1)(x²+x+1)(ax+b)+2(x²+x+1)+3x-2
=(x³-1)(ax+b)+2x²+5x-4

(2)
q1(x)=(x-1)(ax+b)+2
q3(x)=ax+b
→q1(x)=(x-1)q3(x)+2
q1(2)=q3(2)+2
q3(2)-q1(2)=-2