# I don't recognize the form x^18-y^18 and i dont know to factor it.

justaguide | Certified Educator

We have to factor x^18 - y^18.

x^18 - y^18

=> (x^9)^2 - (y^9)^2

=> (x^9 - y^9)(x^9 + y^9)

=> (x^3^3 - y^3^3)(x^3^3 + y^3^3)

=> (x^3 - y^3)(x^6 + y^6 + x^3*y^3)(x^3 + y^3)(x^6 + y^6 - x^3*y^3)

=> (x - y)(x^2 + y^2 + xy)(x^6 + y^6 + x^3*y^3)(x + y)(x^2 + y^2 - x*y)(x^6 + y^6 - x^3*y^3)

The required factored form of x^18 - y^18 = (x - y)(x + y)(x^2 + y^2 - xy)(x^2 + y^2 + xy)(x^6 + y^6 + x^3*y^3)(x^6 + y^6 - x^3*y^3)

giorgiana1976 | Student

You could write the given difference as a difference of 2 squares:

x^18-y^18 = (x^9)^2 - (y^9)^2

We know that the difference of 2 squares yields the product:

a^2 - b^2 = (a-b)(a+b)

(x^9)^2 - (y^9)^2 = (x^9 - y^9)(x^9 + y^9)

But x^9-y^9 may be written as a difference o 2 cubes:

x^9-y^9 = (x^3)^3 - (y^3)^3

We'll note x^3 = a and y^3 = b

As you know, the formula of difference of squares is:

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

We'll substitute a and b  and we'll get:

x^9-y^9 = (x^3 - y^3)(x^6 + (xy)^3 + y^6)

x^9-y^9 = (x-y)(x^2 + xy + y^2)[x^6 + (xy)^3 + y^6]

x^18 - y^18 = (x - y)*(x^2 + xy + y^2)*[x^6 + (xy)^3 + y^6]*(x^9 + y^9)