# Solution to previous Problem of the Week

*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

### Dan

#### Latest posts by Dan (see all)

- 2016 MMC Schedule - November 4, 2015
- 2014 MTAP reviewer for Grade 3 - September 30, 2015
- 2015 MTAP reviewer for 4th year solution part 1 - August 22, 2015

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

I’m also writing to let you know of the cool experience my daughter had checking yuor web blog. She learned several issues, including what it’s like to possess an amazing giving mindset to get most people without difficulty fully understand chosen tortuous issues. You really exceeded people’s expected results. Thank you for churning out such great, safe, educational and also unique tips on the topic.

Thank you of this blog. That’s all I’m able to say. You undoubtedly have made this web web site into an item thats attention opening in addition to critical. You certainly know a fantastic deal of about the niche, youve covered a multitude of bases. Fantastic stuff from this the main internet. All more than again, thank you for the blog.

I am now not positive the place you’re getting your information, however good topic. I needs to spend some time learning more or understanding more. Thanks for great information I used to be in search of this information for my mission.