1. **State the problem:** Factor the polynomial $$\frac{1}{6}x^9 + \frac{5}{6}x^8 - \frac{7}{6}x^7 + \frac{1}{6}x.$$ 2. **Identify common factors:** Each term has a factor of $$\frac{1}{6}x$$. Factor this out first: $$\frac{1}{6}x^9 + \frac{5}{6}x^8 - \frac{7}{6}x^7 + \frac{1}{6}x = \frac{1}{6}x(x^8 + 5x^7 - 7x^6 + 1).$$ 3. **Focus on factoring the polynomial inside the parentheses:** $$x^8 + 5x^7 - 7x^6 + 1.$$ 4. **Try to find rational roots using the Rational Root Theorem:** Possible roots are factors of the constant term (±1) divided by factors of the leading coefficient (1), so possible roots are ±1. 5. **Test $$x=1$$:** $$1^8 + 5(1)^7 - 7(1)^6 + 1 = 1 + 5 - 7 + 1 = 0.$$ So, $$x=1$$ is a root. 6. **Divide the polynomial by $$x-1$$ using synthetic or polynomial division:** Divide $$x^8 + 5x^7 - 7x^6 + 1$$ by $$x-1$$ to get quotient $$x^7 + 6x^6 - x^5 - x^4 - x^3 - x^2 - x - 1.$$ 7. **Check if $$x=1$$ is a root of the quotient:** $$1^7 + 6(1)^6 - 1^5 - 1^4 - 1^3 - 1^2 - 1 - 1 = 1 + 6 - 1 - 1 - 1 - 1 - 1 - 1 = 1.$$ Not zero, so no. 8. **Try to factor the quotient further or use grouping:** Group terms: $$(x^7 + 6x^6) + (-x^5 - x^4) + (-x^3 - x^2) + (-x - 1).$$ Factor each group: $$x^6(x + 6) - x^4(x + 1) - x^2(x + 1) - 1(x + 1).$$ 9. **Notice the last three groups share $$(x+1)$$:** Rewrite: $$x^6(x + 6) - (x^4 + x^2 + 1)(x + 1).$$ 10. **No simple factorization emerges; the polynomial is complicated.** **Final factored form:** $$\frac{1}{6}x(x - 1)(x^7 + 6x^6 - x^5 - x^4 - x^3 - x^2 - x - 1).$$ This is the factored form with the linear factor extracted. Further factorization of the 7th degree polynomial is nontrivial and likely requires numerical methods or advanced techniques.