1. ## polynomials,indices,logarithms

1.
Given the identity
$\displaystyle x^4 + x + 1$ = ($\displaystyle x^2+ A$)($\displaystyle x^2-1$) + 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 $\displaystyle 2^{-x}$$\displaystyle -2(2)^{-(x/2)+1} +3 =0 3. Solve the equation \displaystyle log_{3}(2x-3) = 2 - \displaystyle log_{9}$$\displaystyle (2x-1)^2$

2. Originally Posted by wintersoltice
1.
Given the identity
$\displaystyle x^4 + x + 1$ = ($\displaystyle x^2+ A$)($\displaystyle x^2-1$) + 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 $\displaystyle \frac{x^4 + x + 1}{x+1} = \frac{(x^2 + 1)(x^2 -1) + x -2 }{x+1}$

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

Can you take it form here ?

2.
Solve $\displaystyle 2^{-x}$$\displaystyle -2(2)^{-(x/2)+1} +3 =0 write this as \displaystyle 2^{-x}-4(2)^{-\frac{x}{2}} +3 =0 then let \displaystyle (2)^{-\frac{x}{2}} =u so the equation is \displaystyle u^2 -4u +3 =0 3. Solve the equation \displaystyle log_{3}(2x-3) = 2 - \displaystyle log_{9}$$\displaystyle (2x-1)^2$
use the change of base on the log base 9 and consider writing 2 as $\displaystyle log_{3}(9)$

Bobak