remainder theorem hw help..(again)?

Question

Find the remainder if P(x) = x^10 - 3 is divided by x^2-1

Answer 1

qoutient= X^8+X^6+X^4+X^2+1

remainder= -2

:) i hope i helped :)

Answer 2

There's a simple trick here.
Write x^10 -3 as x^10 -1 -2.
Now x^10 -1 = (x^5-1)(x^5+1).
The first of these factors is divisible by x-1
and the second by x+1, so x^10-1 is divisible by x²-1.
Call the quotient Q. (No need to compute it!)
Then (x^10-3)/(x²-1) = Q -2/(x²-1),
So
x^10 -3 = Q(x²-1) -2
and the remainder is -2.
Sure saves a lot of time if one doesn't have to find Q !!

Answer 3

x^10-3= Q(x)(x+1)(x-1) +ax+b ( remainder is ax+b)
Put x = 1
-2= a+b
Put x=-1
-2=-a+b
summing- 4=2b so b=-2 and a = 0
Remainder is -2

Answer 4

idk