## pratu043 3 years ago Find the condition that x^n + y^n may be divisible by x + y.

1. sauravshakya

IF n is odd

2. pratu043

How do you prove that?

3. Libniz

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

4. pratu043

What do you mean by that?

5. sauravshakya

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

6. sauravshakya

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

7. pratu043

ok

8. pratu043

So you should use factor theorem for this?

9. sauravshakya

yep

10. sauravshakya

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

11. pratu043

ok got that.

12. sauravshakya

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

13. pratu043

ok

14. sauravshakya

which is only possible when n is odd

15. sauravshakya

got it?

16. pratu043

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

17. sauravshakya

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

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