Question

(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?

Answer 1

Hello from MrBillDoesMath!

Answer:   a^6 -1  =  (a-1) ( a^2 + a + 1) (a^3 + 1)

Discussion:

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