Question

( x^{100}=left(x^{2}-3 x+2right) )
This question is from
foom, resonance module
Remainden theorm ( rightarrow )
( x^{100}=g(x) cdotleft(x^{2}-3 x+2right)+R(x) )
whey ( Q(n) ) is nome quatient
( Rightarrow x^{100}=varphi(x) cdot(x-2)(x-1)+a x+b )
Put ( x=1 )
( Rightarrow quad begin{aligned} Rightarrow quad 1 &=0+a+b & Rightarrow a+b=1 end{aligned} )
Put ( n=2 rightarrow )
( Rightarrow quad 2^{100}=0+2 a+b )
( =2 a+b=2^{100}=2left(2^{20}-1right) )
( Rightarrow a+a+b=2^{100} )
( Rightarrow a=2-1 quad therefore b=1-aleft(2^{100}right)=2 )

# 8. If the remainder R(x) = ax + b is obtained by dividing the polynomial x100 by the polynomial x2 - 3x + 2 then (A) a = 2100 - 1 (B) b = 2(299 – 1) (C) b = -2(299 - 1) (D) a = 2100

Solution