Question

How do you factor `x^(4n) - y^(4n)`?

Answer 1

`(x^(2n)+y^(2n))(x^n+y^n)(x^n-y^n)`

Explanation:

Note that both `x^(4n)` and `y^(4n)` are squared terms:

• `x^(4n)=(x^(2n))^2`
• `y^(4n)=(y^(2n))^2`

With this knowledge, we will factor this expression as a difference of squares:

• `color(red)a^2-color(blue)b^2=(color(red)a+color(blue)b)(color(red)a-color(blue)b)`

Thus, we see that

`x^(4n)-y^(4n)`

`=(color(red)(x^(2n)))^2-(color(blue)(y^(2n)))^2`

`=(color(red)(x^(2n))+color(blue)(y^(2n)))(color(red)(x^(2n))-color(blue)(y^(2n)))`

Notice that we can factor `(x^(2n)-y^(2n))` in the same way:

`x^(4n)-y^(4n)=(x^(2n)+y^(2n))(x^(2n)-y^(2n))`

`=(x^(2n)+y^(2n))((x^n)^2-(y^n)^2)`

`=(x^(2n)+y^(2n))(x^n+y^n)(x^n-y^n)`