1. **State the problem:** We need to solve the polynomial equation $$10x^{10} + 111x^{9} - 106x^{8} - 2791x^{7} + 166x^{6} + 12469x^{5} - 774x^{4} - 2809x^{3} + 5424x^{2} - 18180x + 6480 = 0$$ using synthetic division. 2. **Recall synthetic division:** Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form $x - r$. It helps us find roots by testing possible rational roots and simplifying the polynomial. 3. **Step 1: List coefficients:** $$[10, 111, -106, -2791, 166, 12469, -774, -2809, 5424, -18180, 6480]$$ 4. **Step 2: Find possible rational roots:** Using the Rational Root Theorem, possible roots are factors of the constant term 6480 divided by factors of the leading coefficient 10. This is a large set, so we test likely candidates such as $x=1, -1, 2, -2, 3, -3, 4, -4, 5, -5$ etc. 5. **Step 3: Test $x=1$ using synthetic division:** - Bring down 10. - Multiply 10 by 1, add to 111: 121. - Multiply 121 by 1, add to -106: 15. - Multiply 15 by 1, add to -2791: -2776. - Multiply -2776 by 1, add to 166: -2610. - Multiply -2610 by 1, add to 12469: 9859. - Multiply 9859 by 1, add to -774: 9085. - Multiply 9085 by 1, add to -2809: 6276. - Multiply 6276 by 1, add to 5424: 11700. - Multiply 11700 by 1, add to -18180: -6480. - Multiply -6480 by 1, add to 6480: 0. Remainder is 0, so $x=1$ is a root. 6. **Step 4: Write the quotient polynomial:** $$10x^{9} + 121x^{8} + 15x^{7} - 2776x^{6} - 2610x^{5} + 9859x^{4} + 9085x^{3} + 6276x^{2} + 11700x - 6480$$ 7. **Step 5: Repeat synthetic division with the quotient polynomial to find more roots.** For brevity, we show the next root found is $x=2$ (testing similarly). 8. **Step 6: Synthetic division by $x=2$ on the quotient polynomial:** - Coefficients: [10, 121, 15, -2776, -2610, 9859, 9085, 6276, 11700, -6480] - Bring down 10. - Multiply 10 by 2, add to 121: 141. - Multiply 141 by 2, add to 15: 297. - Multiply 297 by 2, add to -2776: -2182. - Multiply -2182 by 2, add to -2610: -6974. - Multiply -6974 by 2, add to 9859: -4089. - Multiply -4089 by 2, add to 9085: 907. - Multiply 907 by 2, add to 6276: 8090. - Multiply 8090 by 2, add to 11700: 27880. - Multiply 27880 by 2, add to -6480: 49280. Remainder is 49280, not zero, so $x=2$ is not a root. 9. **Step 7: Test $x=-1$:** - Bring down 10. - Multiply 10 by -1, add to 121: 111. - Multiply 111 by -1, add to 15: -96. - Multiply -96 by -1, add to -2776: -2680. - Multiply -2680 by -1, add to -2610: 70. - Multiply 70 by -1, add to 9859: 9789. - Multiply 9789 by -1, add to 9085: -704. - Multiply -704 by -1, add to 6276: 6980. - Multiply 6980 by -1, add to 11700: 4720. - Multiply 4720 by -1, add to -6480: -11200. Remainder is -11200, not zero. 10. **Step 8: Test $x=3$:** - Bring down 10. - Multiply 10 by 3, add to 121: 151. - Multiply 151 by 3, add to 15: 468. - Multiply 468 by 3, add to -2776: -1372. - Multiply -1372 by 3, add to -2610: -6726. - Multiply -6726 by 3, add to 9859: -4319. - Multiply -4319 by 3, add to 9085: -2872. - Multiply -2872 by 3, add to 6276: -5400. - Multiply -5400 by 3, add to 11700: -5400. - Multiply -5400 by 3, add to -6480: -22080. Remainder is -22080, not zero. 11. **Step 9: Test $x=4$:** - Bring down 10. - Multiply 10 by 4, add to 121: 161. - Multiply 161 by 4, add to 15: 659. - Multiply 659 by 4, add to -2776: -1040. - Multiply -1040 by 4, add to -2610: -6770. - Multiply -6770 by 4, add to 9859: -2021. - Multiply -2021 by 4, add to 9085: 1001. - Multiply 1001 by 4, add to 6276: 10280. - Multiply 10280 by 4, add to 11700: 52820. - Multiply 52820 by 4, add to -6480: 204240. Remainder is 204240, not zero. 12. **Step 10: Test $x=5$:** - Bring down 10. - Multiply 10 by 5, add to 121: 171. - Multiply 171 by 5, add to 15: 870. - Multiply 870 by 5, add to -2776: 2674. - Multiply 2674 by 5, add to -2610: 10760. - Multiply 10760 by 5, add to 9859: 63959. - Multiply 63959 by 5, add to 9085: 321880. - Multiply 321880 by 5, add to 6276: 161316 - Multiply 161316 by 5, add to 11700: 818580 - Multiply 818580 by 5, add to -6480: 4082400 Remainder is 4082400, not zero. 13. **Step 11: Summary:** We found one root $x=1$ and reduced the polynomial degree by 1. Further roots require more testing or other methods (e.g., factoring, numerical methods). **Final answer:** One root is $x=1$. The polynomial can be factored as $$ (x - 1)(10x^{9} + 121x^{8} + 15x^{7} - 2776x^{6} - 2610x^{5} + 9859x^{4} + 9085x^{3} + 6276x^{2} + 11700x - 6480) = 0 $$ Further roots require additional methods beyond synthetic division shown here.