Factoring x^6 - y^6 as a difference of squares vs cubes?

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Homework Help Overview

The discussion revolves around factoring the expression x6 - y6, specifically exploring its representation as a difference of squares and cubes. Participants are examining the relationships between different forms of the expression and questioning the validity of various factorization methods.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express confusion regarding the factorization of x6 - y6 and its equivalence to other forms like (x2)3 - (y2)3. There are attempts to factor expressions such as x4 + x2y2 + y4, with some questioning the feasibility of these factorizations based on online tools.

Discussion Status

The discussion is active, with participants sharing their thoughts and methods. Some have provided hints and references to online resources, while others express skepticism about the reliability of those resources. There is a notable lack of consensus on the factorization of certain expressions, indicating ongoing exploration of the topic.

Contextual Notes

Participants mention constraints related to homework rules and the challenge of finding valid factorizations. There is also a reference to differing opinions on the capabilities of online factorization tools.

Esoremada
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Homework Statement



Factor x6 - y6

Homework Equations



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

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

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

ehild
 
Esoremada said:

Homework Statement



Factor x6 - y6

Homework Equations



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

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.

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

ehild
 
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?
 
M Quack said:
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?

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
 
Esoremada said:

Homework Statement



Factor x6 - y6


Homework Equations



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

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.

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

RGV
 
Esoremada said:
http://www.freemathhelp.com/factoring-calculator.php

(x^4 + x^2y^2 + y^4)

The polynomial is not factorable with real numbers.
That's what the website says

To paraphrase Groucho Marx: What you gonna' believe, that website or your lyin' eyes?
 
Last edited:
  • #10
I understand now, thanks for the help :smile:

Not going to trust that website as much anymore
 
  • #11
Esoremada said:
How do you factor (x4 + x2y2 + y4)? Online factorers are saying it's not possible.

ehild said:
x4 + x2y2 + y4= (x4 + 2x2y2 + y4) - x2y2

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:
  • #12
Esoremada said:
I understand now, thanks for the help :smile:

Not going to trust that website as much anymore

You should try http://www.wolframalpha.com/
It can handle much more advanced problems.
 
  • #14
rcgldr said:
What ehild has shown is the difference of two squares, not factors. Note the problem statement is asking for the difference of two squares.

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

hild
 
  • #15
ehild said:
@rcgldr - I must not give full solution. It was a hint. The difference of two squares is easy to factorize.
I somehow missed post #6 where you explained this.
 
Last edited:

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