Yahoo Answers: Answers and Comments for F.4 Maths(Polynomial) [數學]
Copyright © Yahoo! Inc. All rights reserved.
https://hk.answers.yahoo.com/question/index?qid=20061129000051KK02601
From Vincent Wong
zhHantHK
Wed, 29 Nov 2006 17:55:25 +0000
3
Yahoo Answers: Answers and Comments for F.4 Maths(Polynomial) [數學]
292
38
https://hk.answers.yahoo.com/question/index?qid=20061129000051KK02601
https://s.yimg.com/zz/combo?images/emaillogohk.png

From tinchoi: When a polynomial g(x) is divided by x1, the ...
https://hk.answers.yahoo.com/question/index?qid=20061129000051KK02601
https://hk.answers.yahoo.com/question/index?qid=20061129000051KK02601
Wed, 29 Nov 2006 18:11:16 +0000
When a polynomial g(x) is divided by x1, the remainder is 5. When g(x) is divided by 2x+1, the remainder is 2.Find the remainder when g(x) is divided by 2x2x1.
By remainder throrem, when g(x) is divided by x1, the remainder
= g(1)
So g(1) = 5 ............. (1)
By remainder throrem, when g(x) is divided by 2x+1, the remainder
= g(1/2)
So g(1/2) = 2 .......... (2)
As the power of 2x²x1 is 2, so the remainder of g(x) divided by 2x²x1 is of max. power of 1.
Let g(x) = (2x²x1)Q + R
Where Q(x) is the quotient and Rx + C is the remainder
g(x) = (2x²x1)Q(x) + Rx + C
g(x) = (x1)(2x+1)Q(x) + Rx + C ........ (*)
Substitute x=1 into (*)
g(1) = (11)(2*1+1)Q + R(1) + C
g(1) = R + C
5 = R + C ............. by (1)
C = 5  R ............. (3)
Substitute x=1/2 into (*)
g(1/2) = (1/21)[2*(1/2)+1]Q + R(1/2) + C
g(1/2) = R/2 + C
2 = R/2 + C ............. by (2)
2 = R/2 + (5R)
4 = R + 2(5R)
4 = R + 10  2R
3R = 6
R = 2
Substitute R=2 into (3)
C = 52
C = 3
So the remainder when g(x) is divided by 2x²x1
= Rx + C
= 2x + 3
20061129 18:16:12 補充：
小小補充：當 g(x) divided by f(x)，remainder 會是 f(x) 的 (maximum power of x)1。所以當 g(x) divided by 2x²x1，remainder 可以假設是 Rx + c。加上題目先給予的兩個提示 g(1) 和 g(1/2) 就可以用兩條方程解兩個 variable。