polynomials,indices,logarithms

**wintersoltice** · September 23rd 2008, 05:18 AM

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 - $log_{9}$ $(2x-1)^2$

**bobak** · September 23rd 2008, 05:50 AM

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