Question

In each pair of polynomials below find what kind of natural number you n must be, so that the first is a factor of the second. 1)x-1,x^n-1

Answer 1

Solution:

For $(x - 1)$
to be a factor of $( {x}^{n} - 1)$
n can be any natural number from

n = 1,2,3,4

$for \: n = 1 = > ( {x}^{1} - 1) = (x - 1) \\ \\ for \: n = 2 = > ( {x}^{2} - 1) = ( x- 1)(x + 1) \\ \\ for \: n =3 \\ \\ ( {x}^{3} - 1) = (x - 1)( {x}^{2} - x + 1) \\ \\ for \: n = 4 = > \\ ( {x}^{4} - 1) = ( {x}^{2} - 1)( {x}^{2} + 1) \\ \\ = (x - 1) (x+ 1)( {x}^{2} + 1)$
So, n can be 1,2,3,4

Hope it helps you.