Ehrenfried Walther von Tschirnhaus (1646-1716) generalizes the linear substitution that eliminates the x^n-1 term in the nth degree polynomial to eliminate the x^n-2 and x^n-3 terms as well. Gottfried Wilhelm Leibniz (1646-1716) had pointed out that trying to get rid of the x^n-4 term usually leads to a harder equation than the original one.

Michael Rolle (1652-1719) proves that f'(x) has an odd number of roots in the interval between two successive roots of f(x).

Edmund Halley (1656-1742) discusses interative solutions of quartics with symbolic coefficients.

Daniel Bernoulli (1700-1782) expresses the largest root of a polynomial as the limit of the ratio of the successive power sums of the roots.

Leonard Euler (1707-1783) tries to find solutions of polynomial equations of degree n as sums of nth roots, but fails.

Halley solves the quadratic in trigonometric functions.

Colin Maclaurin (1698-1746) generalizes Newton's relations for powers greater than the degree of the polynomial.