Question

For which natural numbers `m` is polynomial `(x+1)^m+(x-1)^m` divisible by x?

Answer 1

When `m` is odd.

Explanation:

If `m` is even, we will have `+1` in the expansion of `(x+1)^m` as well as `(x-1)^m` and as `2` appears, it may not be divisible by `x`.

However, if `m` is odd, we will have `+1` in the expansion of `(x+1)^m` and `-1` in the expansion of `(x-1)^m` and they cancel out and as all monomials are various powers of `x`, it will be divisible by `x`.

Answer 2

Odd numbers

Explanation:

Note that the constant term of `(x+1)^m` is `1^m = 1`, whereas the constant term of `(x-1)^m` is `(-1)^m`, which alternates between `-1` for odd values of `m` and `1` for even values of `m`.

So these constant terms cancel out precisely when `m` is odd.

Answer 3

`"for all odd numbers "m`

Explanation:

`"The constant term after expanding with the binomium of"`
`"Newton has to be zero and is equal to : "`

`1^m + (-1)^m = 0`

`=> m" odd because then we have "1-1 = 0.`