How do I find the residue of a function with a huge exponent?

Question

How would I find the remainder of a function that has a huge exponent that would take ages to work out?

Say I have something like this:

$\frac{5x^{110} + x^4 - 7x^2 - 6}{x-1}$

I honestly don't know how to do division like this other than manually decreasing the exponent by $1$ each time until I get to $0$. That would take me ages lol.

score 2 · Answer 1 · answered May 01 '13 at 08:53

2

Using Polynomial remainder theorem,

if $f(x)=5x^{110}+x^4-7x^2-6,$ the remainder will be $f(1)$

answered May 01 '13 at 08:53

lab bhattacharjee

274,582

I don't get that. The wiki page doesn't show a good example for more than an exponent of 3. – Cozen May 01 '13 at 09:21
So essentially the numerator doesn't matter to determine what the value would be? So for this problem, I'd just plug 1 in for x? – Cozen May 01 '13 at 09:23
@Justin, could you follow the proof (http://en.wikipedia.org/wiki/Polynomial_remainder_theorem#Proof)? – lab bhattacharjee May 01 '13 at 09:24
@Justin, numerator needs to be a polynomial – lab bhattacharjee May 01 '13 at 09:26
I tried but it's hard to grasp without number. I think I understand it though. Yeah. So if the numerator is a polynomial and the denominator is in the form (x +- z), I plug the value of z in as x to get the remainder? If the denominator is (x+z), I'd plug -z in and visa versa, right? – Cozen May 01 '13 at 09:31
@Justin, observe that $x^n-a^n$ is divisible by $x-a$ for integer $n\ge1$, So $(x-a)$ divides $f(x)-f(a)$ So, $f(x)=f(x)-f(a)+f(a)=(x-a)q(x)+f(a)$ – lab bhattacharjee May 01 '13 at 09:34
Hmm. I get it up to where you say "so." f(x)=f(x)−f(a)+f(a)=(x−a)q(x)+f(a) isn't clicking =/ – Cozen May 01 '13 at 09:38
let us continue this discussion in chat – lab bhattacharjee May 01 '13 at 10:18

score 0 · Answer 2 · answered May 01 '13 at 08:41

0

In general, the remainder can be found by a bit of modular arithmetic.

$$ 5x^{110} + x^4 - 7x^2 - 6 \equiv 5+1-7-6 \mod x-1 $$ (because $x\equiv 1 \pmod{x-1}$)

answered May 01 '13 at 08:41

Glen O

12,425

Uhh how would I do that on paper? lol – Cozen May 01 '13 at 08:43
Pretty much as written. You'd note that $x\equiv 1\mod{x-1}$, and therefore the remainder on division by $x-1$ is $5+1-7-6=-7$. – Glen O May 01 '13 at 22:16

How do I find the residue of a function with a huge exponent?

2 Answers2