The reminder when x^100 is dividend by x^2-3x+2 is
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x^100 =(x^2 -3x +2)(q(x)) + (ax+b)
x^100 =(x-1)(x-2)(q(x)) + (ax+b)
At x=1
1^100=0*q(1) +a(1)+b
1=a+b (1)
At x=2
2^100=0*q(2) +a(2)+b
2^100=2a+b (2)
Solve (1) and (2) by elimination
2^100 -1= a
1=2^100 -1 +b
b=2-2^100
ax+b=(2^100 -1)x + (2-2^100)
x^100 =(x-1)(x-2)(q(x)) + (ax+b)
At x=1
1^100=0*q(1) +a(1)+b
1=a+b (1)
At x=2
2^100=0*q(2) +a(2)+b
2^100=2a+b (2)
Solve (1) and (2) by elimination
2^100 -1= a
1=2^100 -1 +b
b=2-2^100
ax+b=(2^100 -1)x + (2-2^100)
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