Find the Remainder!!!!!!!!!!!!

Iris

2009-10-25 8:47 pm

1)When the polynomial x^3 - 2x^2 +mx +n is divided by x-1,the remainder is -2.When the polynomial is divided by x+2,the remainder is -26,find the value of m and n.

2)Let f(x )=x^100-1
a)Find the remainder when f(x )is divided by x+1.
b)Using the result of (a ),find the remainder when 2^100 is divided by 3.

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回答 (1)

用戶2053

2009-10-25 11:53 pm

✔ 最佳答案

1)By remainder theorem :

F(1) = 1^3 - 2(1)^2 + m(1) + n = -2
F(-2) = (-2)^3 - 2(-2)^2 + m(-2) + n = -26

F(1) = m+n-1 = -2
F(2) = n - 2m - 16 = -26

F(1) - F(2) :

3m + 15 = 24
m = 3
sub to F(1) : 3+n-1=-2
n = - 4

2a) By remainder theorem :

The remainder = f(-1) = (-1)^100 - 1 =0

b) f(2) = 2^100 - 1

by a) : when f(x) is divided by x+1 ,the remainder = 0, here x = 2

i.e. when f(2) = 2^100 -1 is divided by 2+1 = 3 , the remainder = 0

So 2^100 -1 = 3p , p is a integer.

2^100 = 3p + 1

the remainder is 1

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