Factor the given polynomials:

\(x^{3}-x^{2}+x-1=x^{2}(x-1)+(x-1)\)

\(=x(-1)(x^{2}+1)\)

\(x^{2}-1=(x(-1)(x+1)\)

The lowest common multiple is the product of all the factors of each polynomial to the highest degree. Each of the factors \(x-1\), and \(x^{2}+1\) all have a degree of 1 so the lowest common multiple is \((x-1)(x+1)(x^2+1)\).

\(x^{3}-x^{2}+x-1=x^{2}(x-1)+(x-1)\)

\(=x(-1)(x^{2}+1)\)

\(x^{2}-1=(x(-1)(x+1)\)

The lowest common multiple is the product of all the factors of each polynomial to the highest degree. Each of the factors \(x-1\), and \(x^{2}+1\) all have a degree of 1 so the lowest common multiple is \((x-1)(x+1)(x^2+1)\).