Remainder Theorem

1. Apr 23, 2009

duki

1. The problem statement, all variables and given/known data

Find the remainder when (x^80 - 8x^30 + 9x^24 + 5x + 6) is divided by (x+1)

2. Relevant equations

3. The attempt at a solution

So I'm not really sure where to start. I tried starting by doing long polynomial division, but I get stuck. How do I start this?

2. Apr 23, 2009

rock.freak667

What does the remainder theorem say?

3. Apr 23, 2009

Dick

Yeah, you would get stuck doing the division. It's a long haul. But look, suppose you did do the division f(x)=(x^80 - 8x^30 + 9x^24 + 5x + 6) by (x+1)? Then you would get f(x)=q(x)*(x+1)+r, right? Where q(x) is the quotient and r is the remainder. What happens if you put x=(-1) into that?

4. Apr 23, 2009

duki

Remainder Theorem:
If p(x) / (x – a) = q(x) with remainder r(x),

then p(x) = (x – a) q(x) + r(x).

5. Apr 23, 2009

duki

You get r ?

6. Apr 23, 2009

Dick

Well, yes. You get f(-1)=r. That's the remainder theorem. So what is r?

7. Apr 23, 2009

duki

Do you get:

(1 - 8 + 9 -5 + 6) = 3 ? So r = 3?

8. Apr 23, 2009

Dick

Sure. If you don't believe it make up a simpler example where you can actually do the long division and check that it works. It's good for you.

9. Apr 23, 2009

duki

Thanks. How did you know to use -1?

10. Apr 23, 2009

Dick

Look back at the problem. I'll give you three guesses. The first one had better be right.

11. Apr 23, 2009

duki

Because a = x + 1, so x = -1?

12. Apr 23, 2009

Dick

What's a???? If a=x+1 then x=a-1. You are onto the second guess.

Last edited: Apr 23, 2009
13. Apr 23, 2009

duki

Well heck, I'm not sure. :O
I assume we're using -1 because of something to do with (x+1) = 0 or something?

14. Apr 23, 2009

Dick

Yes, if you had spent all day figuring out the q(x) in f(x)=q(x)*(x+1)+r by doing the horrible division, at the end of it all you could realize that you didn't need to find q(x) at all because if you put x=(-1) the q(x) disappears. That's the remainder theorem.

15. Apr 23, 2009

duki

Sweets.
So if for example I was dividing by (x-4), I would use 4 instead of -1?

16. Apr 23, 2009

Dick

f(x)=q(x)*(x-4)+r. Sure, f(4)=r. You don't need to find q(x) before you know the remainder.

17. Apr 23, 2009

duki

How cool is that.

18. Apr 23, 2009

Way cool.