show that (x-1) is a factor of x10-1 andx11-1

Question

Its solution

Answer 1

Assume that the given 2 expressions would be as follows:

f(x) = (x^10) - 1 ··· Ex.1 and g(x) = (x^11) - 1 ··· Ex.2  

If x=1, Ex.1 and Ex.2 are nonexistent so, x ≠ 1 

To solve these problems, use a standard polynomial form shown below:

a^n - b^n = (a - b)(a^(n-1) + (a^(n-2))b + … +a(b^(n-2)) + b(n-1))  ( n: any pos. integers)

Here, a=x, b=1 so, Ex.1 and Ex.2 can be restated as follows:

f(x) = (x^10) - 1 = (x^10) - (1^10) = (x - 1)(x^9 + x^8 + x^7 + … + x + 1)  (x≠ 1) and

g(x) = (x^11) - 1 = (x^11) - (1^11) = (x - 1)(x^10 + x^10 + x^8 + … + x + 1)  (x≠ 1)

Therefore, (x - 1) is a common factor of (x^10) - 1 and (x^11) -1