# Factorization formula

1. Jul 28, 2007

### ehrenfest

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

Is there like a formula with a name easy to remember of which the following is a specific instance:

x^200 -y^200 = (x-y)(x^199+x^198y+... + y^198*x + y^199)

?

2. Relevant equations

3. The attempt at a solution

2. Jul 28, 2007

### Feldoh

Not that I've ever heard...

$$(x+y)(x-y)(x^2+y^2)(x^4+y^4)(x^4+x^3y+x^{2}y^2+xy^3+y^4)(x^4-x^3y+x^2y^2-xy^3+y^4)(x^8-x^6y^2+x^4y^4-x^2y^6+y^8)(x^{16}-x^{12}y^4+x^8y^8-x^4y^{12}+y^{16})$$
$$(x^{20}+x^{15}y^5+x^{10}y^{10}+x^5y^{15}+y^{20})(x^{20}-x^{15}y^5+x^{10}y^{10}-x^5y^{15}+y^{20})(x^{40}-x^{30}y^{10}+x^{20}y^{20}-x^{10}y^{30}+y^{40})$$
$$(x^{80}-x^{60}y^{20}+x^{40}y^{40}-x^{20}y^{60}+y^{80})$$

Sorry wouldn't fint onto one line...

Maybe to can find a pattern to that mess though. If not, well, it took my calculator about 3 second to find the answer, might want to use one the next time you need to factor that...

Last edited: Jul 28, 2007
3. Jul 28, 2007

### rocomath

which calculator do you have? i been thinking about upgrading to a TI-89 ...

4. Jul 28, 2007

### Feldoh

Ti-89 Titanium

5. Jul 28, 2007

### HallsofIvy

Staff Emeritus
Do you mean
$$x^n- y^n= (x-y)(x^{n-1}+ x^(n-2)y+ /cdot/cdot/cdot+ xy^{n-2}+ y^{n-1}$$?
I've never worried about it having a name!

6. Jul 28, 2007

### ehrenfest

Yes. I just forget identities so easily when I have to remember them by their statement not some label. However, you're right, this is a pretty simple one that is easily verifiable and probably doesn't warrant a name!