If n is an odd integer greater than 3, but not a multiple of 3. Prove that $x^3+x^2+x$ is a factor of ${(x+1)}^n-x^n-1$.

If n is an odd integer but not a multiple of 3, then prove that $xy(x+y)(x^2+y^2+xy)$ is a factor of ${(x+y)}^n-x^n-y^n.$

If n is anodd integer greter than 3 but not a multiple of 3 prove that $[{(x+y)}^n-x^n-y^n]$ is divisible by $xy(x+y)(x^2+xy+y^2).$

If $n$ is n odd integer that is greater than or equal to 3 but not a multiple of 3, then prove that ${(x+1)}^n=x^n-1$ is divisible by $x^3+x^2+x\stackrel{.}{}$

If n is an odd positve integer greater than 3 then $(n^3-n^2)-3(n^2-n)$ is always a multiple of

Find the remainder when ${(x+1)}^n$ is divided by ${(x-1)}^3$. Also show that ${(x-1)}^2$ is a factor of $x^{n+1}-x^n-x+1$ .

If n is an odd integer greater than or equal to 1, then the value of $n^3-{(n-1)}^3+{(n-1)}^3-{(n-1)}^3+....+{(-1)}^{n-1}{1}^3$

If a is defined by f $\left(x\right)=a_0{x}^n+a_1{x}^{n-2}+a_2{x}^{n-2}+...+a_{n-1}x+a_n$ where n is a non negative integer and $a_0,a_1,a_2,\dots \dots .,a_n$ are real numbers and $a_0\ne 0,$ then f is called a polynomial function of degree n. For polynomials we can define the following theorem
(i) Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and 'a' be a real number. if p(x) is divided by (x-a), then the remainder is equal to p(a).
(ii) Factor theorem : Let p(x) be a polynomial of degree greater than or equal to 1 and 'a' be a real number such that p(a) = 0, then (x-a) is a factor of p(x). Conversely, if (x-a) is a factor of p(x). then p(a)=0.
The factor of the polynomial $x^3+3x^2+4x+12$ is

Prove that $x^3+x^2+x$ is factor of ${(x+1)}^n-x^n-1$ where n is odd integer greater than 3, but not a multiple of 3.

Use factor theorem to prove that (x+a) is a factor of $(x^n+a^n)$ for any odd positive integer $n$ .