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

2 Answers
Feb 5, 2018

((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.

Feb 5, 2018

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.