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

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SUMMARY

The discussion focuses on factoring the expression x6 - y6 using the difference of squares and cubes. Participants clarify that x6 - y6 can be expressed as (x2)3 - (y2)3, leading to the factors (x2 - y2)((x2)2 + x2y2 + y22). The expression x4 + x2y2 + y4 is confirmed to be factored as (x2 + xy + y2)(x2 - xy + y2), despite online factorers indicating otherwise. The discussion emphasizes the importance of understanding the difference of squares in polynomial factoring.

PREREQUISITES
  • Understanding of polynomial identities, specifically a3 - b3 and a2 - b2
  • Familiarity with factoring techniques for polynomials
  • Knowledge of the difference of squares identity
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the difference of squares and its applications in polynomial factoring
  • Learn about polynomial identities and their proofs
  • Explore advanced factoring techniques using tools like Wolfram Alpha
  • Practice factoring higher-degree polynomials and recognizing patterns
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Students, educators, and anyone interested in mastering polynomial factoring techniques, particularly in algebra and higher mathematics.

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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