1. **State the problem:** Simplify or analyze the polynomial $x^3 + x^2 - 10x + 8$. 2. **Identify the goal:** We can try to factor the polynomial to find its roots or simplify it. 3. **Use the Rational Root Theorem:** Possible rational roots are factors of the constant term 8 divided by factors of the leading coefficient 1, i.e., $\pm1, \pm2, \pm4, \pm8$. 4. **Test possible roots by substitution:** - For $x=1$: $1^3 + 1^2 - 10(1) + 8 = 1 + 1 - 10 + 8 = 0$ (so $x=1$ is a root). 5. **Divide the polynomial by $(x-1)$ using synthetic or long division:** $$\frac{x^3 + x^2 - 10x + 8}{x - 1} = x^2 + 2x - 8$$ 6. **Factor the quadratic $x^2 + 2x - 8$:** Find two numbers that multiply to $-8$ and add to $2$: these are $4$ and $-2$. 7. **Write the factorization:** $$x^2 + 2x - 8 = (x + 4)(x - 2)$$ 8. **Final factorization of the cubic:** $$x^3 + x^2 - 10x + 8 = (x - 1)(x + 4)(x - 2)$$ 9. **Roots of the polynomial:** $x = 1, -4, 2$. This completes the factorization and root finding of the polynomial.