What is the remainder when x^101-x^4+2 is divided by x+1?(show work)?

Answer 1

The remainder is 0.
To see this write the polynomial as
x^101+ 1 -(x^4 -1)
Since 101 is an odd exponent,
x^101 is divisible by x+1
Also x^4-1 = (x²+1)(x-1)(x+1)
Thus x+1 divides both terms and so divides x^101 -x^4+2.

Answer 2

remainder when f(x) = x^101 --x^4 + 2 is divided by (x + 1)
= f(a) where a is the root of (x+1)=0, a = --1
= f(--1)
= (--1)^101 --(--1)^4 + 2
= --1 --1 + 2
= 0

Answer 3

*This problem is almost impossible to solve without synthetic division

use synthetic division:

-1l 1 -1 2
' ___-1__2__
...' 1 -2 4

Remember that 1 is followed by 100 zeros in the synthetic division, but we don't have to write them down since they don't do anything to the dividend. we still end up with a 1 and the end.


The 4 is the reminder.


Oh, and to understand synthetic division here is a link. It's pretty easy. and really quick.

Answer 4

(x^101-x^4+2):(x+1)=0
x=0

Answer 5

Just do this by method of... Long Division of Polynomials...