polynomials MC難 數學天才求救!?

What is the remainder when x^5－1 is divided by x^2+1
A x+1 B －x－1 C x－1 D －x*3－1
Sol
x^2+1=0 =>x^2=－1
x^5－1
=>(x^2)^2*x－1
=>(－1)^2*x－1
=>x－1
C
f(x)=7x^52－8x^21+15. Find the remainder when f(x+1)f(x－1) is divided by 6x
A 14 B 30 C 70 D 420
Sol
g(x)=f(x+1)f(x－1)=g(x)*(6x)+p
g(0)=f(1)f(－1)=p
p=(7－8+15)*(7+8+15)=14*30=420
B
g(x)=2x^3+mx^2+nx+5 where m and n are positive integers. The factor of g(x) may be
(1) x+5 (2) x－5 (3) 2x+1
Sol
(1)
2*(－5)^3+25m－5n+5=0
－250+25m+5n+5=0
5m－n=49
(2)
2*5^3+25m+5n+5>0
(3)
2*(－1/2)^3+m/4－n/2+5=0
－1/4+m/4－n/2+5=0
－1+m－2n+20=0
2n-m=19
(1)，(3)