Show that x1is a factor of xn 1 for any positive integer nHi

Show that (x-1)is a factor of x^n -1 for any positive integer n.(Hint:Let f(x)=x^n-1 and use the Factor Theorem.

Solution

The factor theorem is a theorem linking factors and zeros of a polynomial. As the Remainder Theorem points out, if we divide a polynomial p(x) by a factor (x – a ) of that polynomial, then we will get a zero remainder. As per the Division Algorithm expression of the polynomial, if: p(x) = (x – a)q(x) + r(x), then, if (x – a) is a factor of p(x), the remainder r (x) after division by (x – a ) will be zero. That is: p(x) = (x – a)q(x). If we synthetic divide a polynomial by x = a and get a zero remainder, then, not only is x = a a zero of the polynomial (by virtue of the Remainder Theorem), but x – a is also a factor of the polynomial (by virtue of the Factor Theorem). For any positive integer n, x = 1 is a zero of the polynomial xⁿ – 1 as ( 1)n = 1 . Thus, by Factor Theorem, ( x – 1) is a factor of the polynomial xⁿ – 1.