When did (cognates of) the word polynomial become standard in mathematical writing for expressions like $x^2 -x + 1$ and $ax^2 + bxy + cy^2$?
In the 3rd edition of an 1822 English translation by Rev. John Hewlett of a French edition of Euler's Elements of Algebra, 1770, one does not see the word polynomial for such expressions---they are usually called "formulas".
Legendre's 1798 Essai sur la Théorie des Nombres uses the word polynôme. Various cognates of this word (e.g., Polynom, polynomial, etc.) appear regularly in publications from then till the present.
The title of Gauss's PhD Thesis of 1799 has the phrase Omnem Functionem Algebraicam Rationalem Integram, which translates into English as "all integral (whole) rational algebraic functions". Gauss continues using similar phrases in his famous Disquitiones Arithmeticae (1801).
Such a phrase appears in many German publications throughout the 19th century as ganze rationale algebraische Funktion, often shortened to ganze rationale Funktion, or even ganze Funktion. It was used by Dedekind in 1857, by Dedekind and Weber in their 1882 paper giving an algebraic treatment of Riemann surfaces, and in Weber's influential Lehrbuch der Algebra (1895, 1896).
Dirichlet's 1840 paper on number theory uses the word polynôme.
George Boole's Mathematical Analysis of Logic, 1847, uses simply function or expression.
Joseph Serret's Cours d'algèbre supérieure, 1849, uses both polynôme and fonction rationale entière.
George Chrystal's Algebra of 1885 uses English translations of Gauss's terminology.
I did not find the word Polynom while browsing through Hilbert's Collected Works. He does use Gauss's wording in some of his publications.
Modern algebra was introduced, in the German language, by the trio of E. Noether (German), E. Artin (Austrian) and B. van der Waerden (Dutch). Noether was the oldest, van der Waerden the youngest.
Noether started out using Gauss's wording in 1908, then used both wordings in two papers in 1915, then just Gauss's wording in two more papers of 1915 and in one paper in 1916. It seems that she permanently switched to Polynom in 1918.
Artin started with a paper in 1924 using Gauss's terminology, but then switched to using Polynom in 4 papers of 1927 (two with Schreier).
Van der Waerden seems to have used only Polynom, starting in 1927, and most importantly in his very influential two volumes (1930, 1931) of Moderne Algebra.
So it seems that the abandonment of Gauss's terminology by the leading German language algebraists was taking hold near the end of WWI. Had the French been responsible for the modern algebra, then of course one would expect the terminology to be polynôme; but why did the German language leaders in algebra choose to go with Polynom instead of Gauss's terminology? Was it connected to the passing of the old guard, Weber (1913) and Dedekind (1916)?
