Question

The remainder when 1-x + x2 - x3 + x4 _ x5 +________+ x1000 is divided by (x+1) is

0
1000
1001
none of these

Answer 1

Step-by-step explanation:

As per the reminder theory for any 

polynomial equation 

p(x)=g(x).q(x)+r(x)→(i)

where p(x),g(x)= polynomials 

q(x)= quotient 

r(x)= reminder 

Assuming r(x)=Ax+B

here p(x)=x100,g(x)=x2−3x+2

Simplifying g(x)

g(x)=x2−3x+2

=x2−2x−x+2

=(x−2)(x−1)

Substituting in eqn (i)

p(x)=(x−2)(x−1).q(x)+Ax+B

taking (x)=1

p(1)=(1−2)(1−1).q(1)+A(1)+B

1100=0+A+B

A+B=1→(i) 

taking x=2

p(2)=(2−2)(2−1).q(x)+A(2)+B

2100=0+2A+B

2A+B=2100→(ii)

from eq

Answer 2

1001 is the answer that is