Polynomial

2007-06-01 12:23 am

Let f(x) be a Polynomial with coefficients.
When f(x) is divided by (x-1)(x-3),the remainer is -2x+5

(a)Find f(1)

(b) When f(x) is divided by(x-1)(x-2) the remainer is kx+8,where k is a real comstant.

(i) Find k
(ii) find the remainer when [f(x)]^2007 is divided by x-2

回答 (1)

用戶16828

2007-06-01 1:07 am

✔ 最佳答案

(a)

f(x) = ( x - 1 )( x - 3 ) Q(x) - 2x + 5

Put x = 1 into f(x)

f(1) = 0 - 2 + 5

f(1) = 3

(b) (i)

f(x)^2007 = ( x - 2 ) Q(x) + r

f(x) = ( x - 1 )( x - 2 ) Q(x) + kx + 8

Put x = 1 into f(x),

Therefore,

f(1) = k + 8 = 3

k = -5

(ii)

f(x) = ( x - 1 )( x - 2 ) Q(x) - 5x + 8

f(x)^2007 = [ ( x - 1 )( x - 2 ) Q(x) - 5x + 8 ]^2007

Put x = 2 into f(x)^2007

f(2)^2007 = (-10)^2007

Therefore,

f(x)^2007 = ( x - 2 )Q(x) + r

f(2)^2007 = (-10)^2007

Therefore, the remainder is - ( 10^2007 )

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