Question

Find the condition that x^n + y^n may be divisible by x + y.

Answer 1

IF n is odd

Answer 2

How do you prove that?

Answer 3

sum of odd roots factor always factor out (x+y)

Answer 4

What do you mean by that?

Answer 5

I know how, PLZ wait sec trying to write it down

Answer 6

Oh got it.... let f(x)=x^n + y^n

Answer 7

ok

Answer 8

So you should use factor theorem for this?

Answer 9

yep

Answer 10

now, when f(x) is divided by (x+y) then
REMAINDER = f(-y)
= (-y)^n + y^n

Answer 11

ok got that.

Answer 12

Now, for x^n + y^n to be divisible by x + y..
remainder = 0
or, (-y)^n + y^n=0

Answer 13

ok

Answer 14

which is only possible when n is odd

Answer 15

got it?

Answer 16

ok so when n is odd, (-y)^n = -y
so (-y)^n + y^n = -y + y = 0
thanks!!

Answer 17

oh its like this:
when n is odd, (-y)^n = -(y)^n
so (-y)^n + y^n = -(y)^n + y^n = 0