2022/005) When $x^{1000}$ is divided by $x^2-4x+3$ what is the remainder?

 When $x^{1000}$ is divided by $x^2-4x + 3$ the raminder shall be a polynomial of degree 1 that is Ax + B

So $x^{1000} = (x^2 - 4x +3) P(x) + Ax + B$ where P(x) is quotient 

We need to find A and B

Now $x^2-4x + 3 = (x-1)(x-3)$

So $x^{1000} = (x-1)(x-3) P(x) + Ax + B$

Putting x = 1 we get $1= A + B\cdots(1)$

Putting x = 3 we get $3^{1000} = 3A + B\cdots(2)$

Subtracting (1) from (2) we get 2A = $3^{1000} -1$

Or $A = \frac{3^{1000}-1}{2}$

Putting in (1) we get $B= 1 - A = \frac{1-3^{1000}}{2}$

So remainder = $\frac{3^{1000}-1}{2}x + \frac{1-3^{1000}}{2}$