Johann Heinrich Lambert (1728-1777) gives series solutions of trinomial equations:

Etienne Bezout (1730-1783) tries to find solutions of polynomial equations of degree n as linear combinations of powers of an nth root of unity, but fails.

Euler tries to find solutions of polynomial equations of degree n as linear combinations of powers of an nth root, but fails.

Joseph Louis Lagrange (1736-1813) expresses the real roots of a polynomial equation in terms of a continued fraction.

Lagrange expands a function as a series in powers of another function and uses this to solve trinomial equations.

Lagrange shows that polynomials of degree five or more cannot be solved by the methods used for quadratics, cubics, and quartics. He introduces the Lagrange resolvent, an equation of degree n!.

Euler gives series solutions of:

John Rowning (1699-1771) develops the first mechanical device for solving polynomial equations. Although the machine works for any degree in theory, it was only practical for quadratics.