Prove that the polynomial n*x^(n+2)-(n+1)*x^(n+1)+x is divisible by (x-1)^2.

To prove that the polynomial is divisible by (x-1)^2, that
means that x = 1 is a root of polynomial and it's first
derivative.

For this reason, we'll substitute x by 1 in the
expresison of polynomial:

P(1) =
n*1^(n+2)-(n+1)*1^(n+1)+1

Since 1 raised to any power,
yields 1, we'll get:

P(1) = n - (n+1) +
1

We'll remove the
brackets:

P(1) = n - n - 1 +
1

P(1) = 0

So, x = 1 is the
root of the polynomial.

Let's verify if x = 1 is the root
of the 1st derivative.

P'(x) = n*(n+2)*x^(n+1) -
(n+1)^2*x^n + 1

We'll substitute x by 1 in the expresison
of the 1st derivative:

P'(1) = n(n+2) - (n+1)^2 +
1

We'll remove the brackets and we'll raise to square the
binomial:

P'(1) = n^2 + 2n - n^2 - 2n - 1 +
1

We'll eliminate like terms and we'll
get:

P'(1) = 0

Therefore, x =1
is the root of the first derivative,
too.

Then, the root x=1 has the order of
multiplicity of 2, and the polynomial is divisible by
(x-1)^2.