1. Statement of the problem: Factor the polynomial $x^2-1$ and find its roots. 2. Recognize that this is a difference of squares because $x^2-1^2=x^2-1$. 3. Apply the difference of squares formula $a^2-b^2=(a-b)(a+b)$ with $a=x$ and $b=1$. 4. Therefore $$x^2-1=(x-1)(x+1)$$ 5. To find the roots, set each factor equal to zero. 6. From $x-1=0$ we get $x=1$. 7. From $x+1=0$ we get $x=-1$. 8. Check by expanding the factors: $$ (x-1)(x+1)=x^2-1$$ 9. Final answer: The factorization is $(x-1)(x+1)$ and the roots are $x=1$ and $x=-1$.