Question

The remainder when x^4-y^4 is devided by x+y

Answer 1

use algebric formula
(a^2-b^2)=(a-b)(a+b)

now,
x^4-y^4=(x^2-y^2)(x^2+y^2)
=(x-y)(x+y)(x^2+y^2)

now divided by (x+y)

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

Answer 2

Apply difference of squares rule:
= x^{2} - y^2
= (x+y) (x-y) x^4 - y^4
= (x^2+y^2) (x^2-y^2) (x^2+y^2) (x^2-y^2)

Factor:
(x^2-y^2)
= (x+y) (x-y)
= (x + y) (x - y)
= (x + y) (x - y) (x^2+y^2)
=  (x + y) (x - y) (x^2+y^2) / x + y

Cancelling common factor: x + y 

$(x - y) (x^2+y^2)$