Prove x^n - y^n has a factor of (x-y)

**Timbuk2**

Basically for all n >= 1, show that [latex]x^n - y^n[/latex] has a factor of (x-y).

I'm not really sure what to do in this case. I'm guessing that induction is necessary but I'm not sure.

What I did was a throwback to the factor theorem we did at Leaving Cert, and said that

If x-a is a factor of the polynomial f(x), then f(a) = 0 (as a-a is zero), so a is a root.

Similarly, if I say f(x) = [latex]x^n - y^n[/latex], I could say that f(y) = 0.

Does this prove that x-y is a factor? Am I even on the right road in terms of solving this?

seandoiler

i'd prove it using induction to be honest

clearly [latex]x-y[/latex] divides [latex]x^1 - y^1[/latex], assume that [latex]x - y[/latex] divides [latex]x^k - y^k[/latex] for some k, then consider [latex]x^{k+1} - y^{k+1}[/latex] and write this as [latex]x^{k+1} - y^{k+1}=x^{k+1}-xy^{k}+xy^{k}-y^{k+1}[/latex] and factorise to see that [latex]x- y[/latex] divides [latex]x^{k+1} - y^{k+1}[/latex] and result follows

CJC86

seandoiler wrote: »

i'd prove it using induction to be honest

...then consider [latex]x^{k+1} - y^{k+1}[/latex] and write this as [latex]x^{k+1} - y^{k+1}=x^{k+1}-xy^{k}+xy^{k}-y^{k+1}[/latex] and factorise to see that [latex]x- y[/latex] divides [latex]x^{k+1} - y^{k+1}[/latex] and result follows

I hope I'm not missing something obvious, but I'm pretty sure that doesn't help.

I don't think I'd use induction personally, since [latex]x^{k+1}-y^{k+1} = (x-y)(x^k + x^{k-1}y + ... + xy^{k-1} + y^k) [/latex], the second factor of which looks nothing like [latex]x^k-y^k[/latex].

I don't like the look of it, but I think it's fair enough to fix y and use the factor rule from leaving cert. People don't usually use results from one variable when dealing with 2 or more variables, but I think it's ok in this case.

Or, if you know/suspect the second factor which I've written above then you can show it pretty clearly just by multplying it by x and y seperately and then taking the difference.

Having a brain fart atm and can't think of other ways to do this...

LeixlipRed

There's nothing wrong with seandolier's proof. You can rewrite as:

[latex]x^{k+1} - y^{k+1}=x^{k+1}-xy^{k}+xy^{k}-y^{k+1}=x(x^{k}-y^{k}) + y^{k}(x-y)[/latex] which is clearly divisible by [latex]x-y[/latex].

CJC86

Ah yes, just me being a moron so. Apologies.

LeixlipRed

I can't get the bloody plus sign to appear in the last expression there. Grrrr!!! And now the other plus is gone too, ahh. Anyway, you know what I mean

**Timbuk2**

Thanks guys that's great! Your solutions really helped me!

I see now that my attempt isn't very logical, as it assumes that the second variable is fixed. All of your solutions are a lot more direct!

Thanks again!

seandoiler

CJC86 wrote: »

I hope I'm not missing something obvious, but I'm pretty sure that doesn't help.

sorry i probably should have written it out explicitly but was on my way out...thanks leixlipred for finishing it off

Eliot Rosewater

LeixlipRed wrote: »

I can't get the bloody plus sign to appear in the last expression there. Grrrr!!! And now the other plus is gone too, ahh. Anyway, you know what I mean

I think it's something to do with editing posts; vBulletin automatically cuts out the "+" or something.

LeixlipRed

Haha, explains why the +s disappeared sequentially so

MathsManiac

You could of course prove it by just producing the other factor and showing that their product is x^n - y^n. The other factor is (and sorry i'm not a latexer):

Sum from i = 1 to n of [x^(n-i).y^(i-1)]

If you multiply it by x-y you have a telescoping sum that easily simplifies to x^n - y^n.