How do you factor x^9 - x^6 - x^3 + 1?

1 Answer

(x-1)^2(x^2+x+1)^2(x+1)(x^2-x+1)

Explanation:

Start from the given:

x^9-x^6-x^3+1

by grouping method

first two terms, factor x^6 and last two terms, factor the -1

that is

x^6(x^3-1)-1(x^3-1)

factor out the common binomial factor (x^3-1) so that

(x^3-1)(x^6-1)

at this point, use " sum or difference of two cubes" forms
and difference of two squares

a^3-b^3=(a-b)(a^2+ab+b^2)
a^3+b^3=(a+b)(a^2-ab+b^2)
a^2-b^2=(a-b)(a+b)
so that

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

(x-1)(x^2+x+1)(x-1)(x^2+x+1)(x+1)(x^2-x+1)

(x-1)^2(x^2+x+1)^2(x+1)(x^2-x+1)

have a nice day ! from the Philippines...