(a^6-1)

. how would you factor this? difference of squares or.difference of cubes?... this was a question that came up on my homework, and i did difference of squares. doung it on my graphing calculator difference of squares worked and difference of cubes didnt, but i wanna know other opinions on the matter bc my teacher isnt very good and she just ignored me when i suggested testing it with a calculator...

it does come out as different ending equations too
so what would y'all say would be like a rule to use to factor this?

Let's factor a^6 - 1. First, the difference of two squares, such as " b^2 - c^2 factors like this ( b + c) * (b-c). Your expression is the difference of two squares (where the term being squared is a^3)

a^6 - 1 = ( a^3 -1) (a^3 + 1) (Multiply it out!)

But a= 1 is a factor of a^3 - 1 as 1^3 -1 - 0. So we can further factor a^3 - 1. Dividing a^3 -1 by a- 1 gives a^2 + a + 1.

a^3 = (a-1 ) ( a^2 + a + 1).

You may want to multiply the expressions to verify my statements. I did the division but it's not easy to show them on Brainly.

Conclusion:

a^6 -1 = (a^3 + 1) (a^3 -1 ) =

(a^3 + 1) ( (a-1) (a^2 + a + 1 ) )

Thank you,

MrB