1. **State the problem:** Find the possible rational zeros of the polynomial $$f(x) = x^4 - 5x^3 + 5x^2 + 5x - 6$$ and test them to find actual zeros. 2. **Factors of the constant term (p-values):** The constant term is $$-6$$. Factors of $$-6$$ are $$\pm 1, \pm 2, \pm 3, \pm 6$$. 3. **Factors of the leading coefficient (q-values):** The leading coefficient is $$1$$. Factors of $$1$$ are $$\pm 1$$. 4. **Possible rational zeros (± p/q values):** Using the Rational Root Theorem, possible rational zeros are $$\pm \frac{p}{q} = \pm 1, \pm 2, \pm 3, \pm 6$$. 5. **Simplified list of possible rational zeros:** $$\pm 1, \pm 2, \pm 3, \pm 6$$. 6. **Test zeros by substitution:** - Test $$x=1$$: $$f(1) = 1 - 5 + 5 + 5 - 6 = 0$$ So, $$x=1$$ is a root. - Test $$x=-1$$: $$f(-1) = 1 + 5 + 5 - 5 - 6 = 0$$ So, $$x=-1$$ is a root. - Test $$x=2$$: $$f(2) = 16 - 40 + 20 + 10 - 6 = 0$$ So, $$x=2$$ is a root. - Test $$x=-2$$: $$f(-2) = 16 + 40 + 20 - 10 - 6 = 60 \neq 0$$ Not a root. - Test $$x=3$$: $$f(3) = 81 - 135 + 45 + 15 - 6 = 0$$ So, $$x=3$$ is a root. - Test $$x=-3$$: $$f(-3) = 81 + 135 + 45 - 15 - 6 = 240 \neq 0$$ Not a root. - Test $$x=6$$: $$f(6) = 1296 - 1080 + 180 + 30 - 6 = 420 \neq 0$$ Not a root. - Test $$x=-6$$: $$f(-6) = 1296 + 1080 + 180 - 30 - 6 = 2520 \neq 0$$ Not a root. 7. **Conclusion:** The rational roots are $$x = 1, -1, 2, 3$$. 8. **Factorization:** Since these are roots, the polynomial factors as $$f(x) = (x - 1)(x + 1)(x - 2)(x - 3)$$. **Final answer:** The polynomial factors as $$f(x) = (x - 1)(x + 1)(x - 2)(x - 3)$$ and the rational zeros are $$\boxed{\pm 1, 2, 3}$$.