# Homework Help: Factoring x^6 - y^6 as a difference of squares vs cubes?

1. Oct 4, 2012

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

Factor x6 - y6

2. Relevant equations

a3 - b3 = (a - b)(a2 + ab + b2)
a2 - b2 = (a + b)(a - b)

3. The attempt at a solution

I'm confused.

x6 - y6 = (x2)3 - (y2)3 = (x3)2 - (y3)2

So shouldn't they all have the same factors? When I factor (x2)3 - (y2)3 = (x3)2 - (y3)2 I get different results.

(x3)2 - (y3)2
(x3 - y3)(x3 + y3)
(x - y)(x2 + xy + y2)(x + y)(x2 - xy + y2)

(x2)3 - (y2)3
(x2 - y2)((x2)2 + x2y2 + (y2)2)
(x – y)(x + y)(x4 + x2y2 + y4)

How do you factor (x4 + x2y2 + y4) into (x2 + xy + y2)(x2 - xy + y2)? Online factorers are saying it's not possible.

2. Oct 4, 2012

### ehild

x4 + x2y2 + y4= (x4 + 2x2y2 + y4)-x2y2

ehild

3. Oct 4, 2012

### Mentallic

A little something I also discovered for myself back in high school. I assumed it was a closely guarded secret that only exceptional Mathematicians discover in their lifetimes

4. Oct 4, 2012

### ehild

You are an exceptional Mathematician then... But our teacher told us in the school that adding and subtracting the same thing does not hurt.

ehild

5. Oct 4, 2012

### M Quack

Well, if you work out the product (x^2 + xy + y^2)(x^2 -xy + y^2) you get the right result. What else do you want?

6. Oct 4, 2012

### ehild

To find out how to do the factoring if you do not know that these are the factors.

As I wrote before, x4 + x2y2 + y4= (x4 + 2x2y2 + y4)-x2y2=(x2+y2)2-(xy)2.
Apply the identity a2-b2=(a-b)(a+b)

ehild

7. Oct 4, 2012

### Ray Vickson

What online factorers are you using? Certainly Maple can do it with no problem (although it is not an on-line package_.

RGV

8. Oct 4, 2012

9. Oct 4, 2012

### LCKurtz

Last edited: Oct 4, 2012
10. Oct 4, 2012

I understand now, thanks for the help

Not gonna trust that website as much anymore

11. Oct 4, 2012

### rcgldr

What ehild has shown is the difference of two squares, not factors. Note the problem statement is asking for the difference of two squares.

Last edited: Oct 4, 2012
12. Oct 4, 2012

### Mentallic

You should try http://www.wolframalpha.com/
It can handle much more advanced problems.

13. Oct 4, 2012

### D H

Staff Emeritus
14. Oct 4, 2012

### ehild

@rcgldr
I must not give full solution. It was a hint. The difference of two squares is easy to factorize.

hild

15. Oct 5, 2012

### rcgldr

I somehow missed post #6 where you explained this.

Last edited: Oct 5, 2012