I know that in the "Earliest known uses" website the origin is traced back to Viete in the 16th/17th century. The given source, (Cajori 1919, page 139) which is Florian Cajori (1919): "A history of Mathematics" (available at archive.org) is correct, however, nothing is substantiated about the context.

* Does someone have access to the 1591 text to confirm the correctness of the Cajori reference?

Around 1585, Stevins in a french text uses "multinom*" as umbrella notion for "binom*" and "trinom*" in the glossary. These are used exclusively in the meaning "sum of algebraic terms", as found inside a root or power. It can be guessed that Viete knew about the text of Stevin and altered "multinom*" to "polynom*" for essentially the same usage, roots and powers of aggregated algebraic terms.

*Can anyone shed light on the question if Viete used "polynomial" outside of a glossary or how widespread the use of "polynom*" was in texts before the 19th century?

At some time in the late 18th or early 19th century, the "polynomial" began to be used in the modern sense. Perhaps via multinomial and polynomial coefficients. In algebra, as relates to the fundamental theorem of algebra, "polynomial" was not or rarely used until the end of the 19th century. Instead, "entire (rational) (algebraic) function" or just "function" or "equation" with an indication of the degree is used. For instance, Weierstrass in 1891 still used "entire (rational) function" ("ganze (rationale) Function") for polynomials.

* Is there one document that started the shift in usage, or was it gradual?

* As a related question, when did "entire function" stop to indicate a polynomial and start to indicate a power series with infinite convergence radius?