1. polynomials,indices,logarithms

1.
Given the identity
$x^4 + x + 1$ = ( $x^2+ A$)( $x^2-1$) + Bx + C

determine the numerical value of A ,B and C. By giving x a suitable value, find the remainder when 100000101 is divided by 101.

I know how to do the first part, A is 1, B is 1 and C is 2, but don't know how to do part 2 which is in blue colour.

2.
Solve $2^{-x}$ $-2(2)^{-(x/2)+1}$ +3 =0

3.
Solve the equation
$log_{3}$(2x-3) = 2 - $log_{9}$ $(2x-1)^2$

2. Originally Posted by wintersoltice
1.
Given the identity
$x^4 + x + 1$ = ( $x^2+ A$)( $x^2-1$) + Bx + C

determine the numerical value of A ,B and C. By giving x a suitable value, find the remainder when 100000101 is divided by 101.

I know how to do the first part, A is 1, B is 1 and C is 2, but don't know how to do part 2 which is in blue colour.
It should be fairly obvious that the value of x you choose is x = 100 , and your diving by x+1, your values of A B and C are correct.leave the numbers out for now as it makes things simpler.

start with $\frac{x^4 + x + 1}{x+1} = \frac{(x^2 + 1)(x^2 -1) + x -2 }{x+1}$

$\frac{(x^2 + 1)(x+1)(x-1) + x -2 }{x+1} = (x^2 + 1)(x-1) + \frac{x-2}{x+1}$

Can you take it form here ?

2.
Solve $2^{-x}$ $-2(2)^{-(x/2)+1}$ +3 =0
write this as $2^{-x}-4(2)^{-\frac{x}{2}} +3 =0$
then let $(2)^{-\frac{x}{2}} =u$
so the equation is $u^2 -4u +3 =0$

3.
Solve the equation
$log_{3}$(2x-3) = 2 - $log_{9}$ $(2x-1)^2$
use the change of base on the log base 9 and consider writing 2 as $log_{3}(9)$

Bobak