I need the equation for an algebraic answer

Question

X^{6 _} y⁶

Answer 1

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Answer 2

Mariann,

 

Do you need to factor this polynomial? Here is how to do it.

 

You can recognize this as a difference of squares or a difference of cubes:

 

x⁶ − y⁶ = (x²)³ − (y²)³ or (x³)² − (y³)².

 

I will show you how to do this both ways, because it is interesting.

 

Here are the formulas for factoring you will need:

 

a³ − b³ = (a − b)(a² + ab + b²)

a³ + b³ = (a + b)(a² − ab + b²)

a² − b² = (a − b)(a + b)

 

x⁶ − y⁶ = (x²)³ − (y²)³ = (x² − y²)(x⁴ + x²y² + y⁴) = (x − y)(x + y)(x⁴ + x²y² + y⁴)

 

and 

 

x⁶ − y⁶ = (x³)² − (y³)² = (x³ − y³)(x³ + y³) = (x − y)(x² + xy + y²)(x + y)(x² - xy + y²).

 

These answers are both correct, but the second one is completely factored, while the first one is not. If you compare the two, you can notice that 

 

x⁴ + x²y² + y⁴ = (x² + xy + y²)(x² - xy + y²).

 

This is extremely hard to factor by yourself, so the best way to approach a problem like this one is to factor the original polynomial as a difference of squares. This is one of those questions where there are two ways of looking at it, but one of them wins out as getting the answer with less pain and suffering, and we all want that.

 

I hope this helps you, although it is quite a lot of explanation. Feel free to post a reply if you need more help with it.