Question

How do you factor `p^6 - 1`?

Answer 1

`(p+1)(p-1)(p^2-p+1)(p^2+p+1)`

Explanation:

Identities to recognize: `{((a^2-b^2)=(a+b)(a-b)),((a^3+b^3)=(a+b)(a^2-ab+b^2)),((a^3-b^3)=(a-b)(a^2+ab+b^2)):}`

Through the first identity, a difference of squares, `p^6-1=(p^3+1)(p^3-1)`.

Now, we have the other two identities—a sum of cubes and a difference of cubes.

`color(blue)((p^3+1))color(magenta)((p^3-1))=color(blue)((p+1)(p^2-p+1))color(magenta)((p-1)(p^2+p+1))`

Thus, `p^6-1=(p+1)(p-1)(p^2-p+1)(p^2+p+1)`.