An Embellished History of Polynomial Factoring

\(\displaystyle a_1x + a_0 = 0\)
\(\displaystyle a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{11}x^{11} + a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{12}x^{12} + a_{11}x^{11} + a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{13}x^{13} + a_{12}x^{12} + a_{11}x^{11} + a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{14}x^{14} + a_{13}x^{13} + a_{12}x^{12} + a_{11}x^{11} + a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{15}x^{15} + a_{14}x^{14} + a_{13}x^{13} + a_{12}x^{12} + a_{11}x^{11} + a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)
\(\displaystyle a_{16}x^{16} + a_{15}x^{15} + a_{14}x^{14} + a_{13}x^{13} + a_{12}x^{12} + a_{11}x^{11} + a_{10}x^{10} + a_9x^9 + a_8x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\)