Yahoo Answers: Answers and Comments for Two questions on polynomial (F.4) [數學]
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Tue, 17 Apr 2007 23:33:04 +0000
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Yahoo Answers: Answers and Comments for Two questions on polynomial (F.4) [數學]
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From Yin Fai: 1)
Assume the remainder be Ax + B where A and...
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https://hk.answers.yahoo.com/question/index?qid=20070417000051KK05377
Wed, 18 Apr 2007 00:18:46 +0000
1)
Assume the remainder be Ax + B where A and B are constants.
i.e. f(x) = Q (x5)(x+2) + (Ax + B)
As given, f(5) = 9 and f(2) = 5
f(5) = 5A + B = 9  (1)
f(2) = 2A + B = 5  (2)
(1)  (2)
7A = 14
A = 2
Put it back into (1)
5(2) + B = 9
B = 1
So, remainder is 2x  1
**********************
2)
a) Assume f(x) = Q(x+1)(x3) + (2x4) where Q is the quotient.
f(1) = 2(1)  4 = 6
f(3) = 2(3)  4 = 2
b) f(x) = 3x^3 + mx^2  nx  7
f(1) = 3(1)^3 + m(1)^2  n(1)  7 = 6
3 + m + n  7 = 6
m + n = 4  (1)
f(3) = 3(3)^3 + m(3)^2  n(3)  7 = 2
81 + 9m  3n  7 = 2
3m  n = 24  (2)
c)
(1) + (2)
4m = 20
m = 5
n = 9

From myisland8132: 1.When a polynomial f(x) is divided by x5,the...
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https://hk.answers.yahoo.com/question/index?qid=20070417000051KK05377
Wed, 18 Apr 2007 00:04:05 +0000
1.When a polynomial f(x) is divided by x5,the remainder is 9.When it is divided by x+2,the remainder is 5.Find the remainder when f(x) is divided by (x5)(x+2).
SOLUTION
let the remainder is ax+b
then
f(x)=g(x)(x5)(x+2)+(ax+b)
Then by remainder theorem
f(5)=5a+b=9...(1)
f(2)=2a+b=5...(2)
(1)(2):
7a=14
a=2
sub into (2) b=2a5=1
So the remainder when f(x) is divided by (x5)(x+2) is
2x1
2.Let f(x) =3x^3 +mx^2 nx 7 .When f(x) is divided by (x+1)(x3),the remainder is 2x4.
(a)Find the values of f(1) and f(3).
(b)Set up two equations connecting m and n.
(c)Find the values of m and n.
SOLUTION
(a)
We have
f(x)=g(x)(x+1)(x3)+2x4
f(1)=2(1)4=6
f(3)=2(3)4=2
(b)
Since f(1)=6
f(1)=3(1)^3 +m(1)^2 n(1) 7=6
3+m+n7=6
m+n=4...(1)
This is the first equation
Since f(3)=2
f(3)=3(3)^3 +m(3)^2 n(3) 7=2
81+9m3n7=2
9m3n=72
3mn=24
n3m=24...(2)
This is the second equation
(c)
from (1)
m=4n
sub into (2)
n3(4n)=24
12+4n=24
n=9
m=49=5
20070418 00:33:48 補充：
答題中的 g(x) 就是 quotinet.應該用Q(x) 會好看些﹐不過其實都是同一樣東西。