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

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 (x)=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),…….,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 .