Question

The remainder R obtained by dividing x100 by x²-3x+2 is a polynomial of degree less than 2. then r may be written as ?

Answer 1

 Let  p ( x ) =  x^100  and  q ( x ) =  x^2  – 3x  + 2 = ( x  – 1) ( x  – 2)
When p(x) is divided by q(x), then by division algorithm there exists Q(x) and R(x) such that
  p(x)=q(x)+r(x)
x^100=(x-1)(x-2)q(x)+(Ax+B)
put x=1 and x=2
1=A+B and 2^100=2A+B
by solving
A=2^100-1
B=2-2^100
then rem will be...
(2^100-1)x+(2-2^100)

Answer 2

Answer:

(2^100-1)x+(2-2^100)

Step-by-step explanation:

Let  p ( x ) =  x^100  and  q ( x ) =  x^2  – 3x  + 2 = ( x  – 1) ( x  – 2)

When p(x) is divided by q(x), then by division algorithm there exists Q(x) and R(x) such that

 p(x)=q(x)+r(x)

x^100=(x-1)(x-2)q(x)+(Ax+B)

put x=1 and x=2

1=A+B and 2^100=2A+B

by solving

A=2^100-1

B=2-2^100

then remainder will be (2^100-1)x+(2-2^100)