Question

Prove by M.I. that (x+y)^n - x^n - y^n is divisible by xy(x+y).
i am stuck in xyN(x+y)^2 + yx^k + xy^k
what can i do next?
thx.

Answer 1

or is there a simple and quick method to do this?

Answer 2

you sure thats right ? i guess n=2 doesnt work..

Answer 3

@shubhamsrg
the question is actually just to prove that (x+y)^n - x^n - y^n is divisible by xy(x+y)..(by factor theorem) but i thought it may also be prove by M.I. so i wanted to try. We can't do it by M.I.??

Answer 4

I don't think you can with induction. induction is with single variable I think. I have never done it with more than one variable, note we need to show its true for n = 1. how would you do this? I could be wrong...

Answer 5

also, induction only shows when the independent variable is a natural.

Answer 6

when n = 1,
(x+y) - x - y = 0 which is divisible by xy(x+y)
can it be like this?

Answer 7

ahh yeah. But do you need to show this is true for all numbers or just the natural numbers?

Answer 8

for natural numbers

Answer 9

there are some M.I. problems in my book that include both x and y
eg prove x^n - y^n is divisible by x-y

Answer 10

cool cool, I'm playing with it on paper....hope i can see the "trick"

Answer 11

thanks :)

Answer 12

hmm tricky... i cant figure out where to use the hypothesis...

Answer 13

yea..i deliberately make it as (x+y) [(x+y)^k + x^k + y^k] - x^(k+1) - y^(k+1)
but i ended up with xyN(x+y)^2 + yx^k + xy^k... dont know what to do next...

Answer 14

I think WHEN n is even then (x+y)^n - x^n - y^n is not divisible by xy(x+y)...