*What is the remainder when f(x)=999x^{999}+998x^{998}+997x^{997}+. . .+2x^{2}+x is divided by x-1?*

This is a finite polynomial of degree 999. For us to solve this problem we need to know that f(a) is the remainder when f(x) is divided by x-a. That is the famous remainder theorem.

The remainder when* f(x)=999x^{999}+998x^{998}+997x^{997}+. . .+2x^{2}+x* is divided by

**is also**

*x-1**.*

**f(1)****f(1)= 999(1) ^{999}+998(1)^{998}+997(1)^{997}+. . .+2(1)^{2}+(1)**

**f(1)= 999+998+997+. . . +2+1**

Using the formula for the sum of arithmetic series we have,

where * a_{1}* and

**are the first and last term respectively,**

*a*_{n}*is the number of terms.*

**n***f(1)=999(1+999)/2*

*f(1)=499500*

Here is the link to that page

prove the following.in THEORY OF EQUATION

1. If R; +,∙ ›is a ring , then for any a Є R,0 ∙a = a ∙0=0, where 0 is the additive identity.

2. Let R be an abelian ring, then Øc : R[x]→R such that Øc(p(x)) = p(c) is a ring homomorphism

Can you help me to answer this question.Thank you