兩題F4 數

1)
f(x) is divided by x-1,the remainder is 4
f(1)=4...(1)
it is divided by x+1,the remainder is 2
f(-1)=2...(2)
Let f(x)=(x+1)(x-1)Q(x)+(ax+b), where Q(x) is some polynomial
From (1), sub x=1
we have a+b=4...(*)
From (2), sub x=-1
we have -a+b=2...(**)
On solving, a=1 and b=3
So, the remainder is x+3

2)
Let f(x)=(x+1)^2008+1
When f(x) is divided by x, the remainder is f(0)=2
So, by putting x=7 into f(x), the remainder of which 8^2008+1 is divided by 7 is 2
Hence, if today is Saturday, the day after(8^2008+1) days is Monday.

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