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

[Esoremada]

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.

 

[ehild]

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

 

[Mentallic]

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

 

[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

 

[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?

 

[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?

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

 

[Ray Vickson]

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]

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

 

[LCKurtz]

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?

 

[Esoremada]

I understand now, thanks for the help

Not going to trust that website as much anymore

 

[rcgldr]

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

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.

 

[Mentallic]

I understand now, thanks for the help

Not going to trust that website as much anymore

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

 

[D H]

You should try http://www.wolframalpha.com/

http://www.wolframalpha.com/input/?i=factor+x^4+++x^2y^2+++y^4

 

[ehild]

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

 

[rcgldr]

@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.