Question

The remainder when x^(2011) is divided by x^2 -3x+2 is ?

Answer 1

`((2^2011 - 1)x - (2^2011 - 2))/(x^2 - 3x + 2)`

Explanation:

A semi-easy way to see this is to start dividing the expression using Long Division. Write the dividend (under the division symbol) with zeros as
`x^2011 + 0x^2010 + 0x^2009 + 0x^2008 + .... 0`
We won't need all of the terms in order to notice the pattern.
As you start dividing, you will observe that the first term has a coefficient of 1, the second has a coefficient of 3, the third has a coefficient of 7, then 15, then 31, etc..
These numbers have the form `2^m - 1`.
The remainder will appear after you have divided through the whole thing, consisting of the `2011^(th)` and `2012^(th)` terms.

The first term in the quotient will follow the same pattern, having `2^2011-1` as its coefficient. The last coefficient is one less than `2^2011-1` -- it is `2^2011 - 2`, or `2(2^2010 - 1)`.

The same pattern is true for every division of the form
`x^m/(x^2 - 3x + 2)`, where `m >= 3`.

You may also notice that `x^2011 - 1` is a multiple of `x - 1`, which would cancel a factor in the denominator.

Answer 2

See below.

Explanation:

`x^2011 = Q(x)(x-1)(x-2) + a x + b`

where `Q(x)` is a `2009` degree polynomial and `(x-1)(x-2) = x^2-3x+2`

Now we know

`1^2011 = a+b`
`2^2011 = 2a+b`

Solving for `a,b` we obtain

`a = 2^2011-1, b = 2-2^2011` and then

`r(x) = (2^2011-1)x+2-2^2011` which is the remainder.