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Leo Hendrik Baekeland

The first plastic based on a synthetic polymer was made from phenol and formaldehyde, with the first viable and cheap synthesis methods invented in 1907, by Leo Hendrik Baekeland, a Belgian-born American living in New York state. Source: https://en.m.wikipedia.org/wiki/Plastic

In 1856 Alexander Parkes patented "Parkesine" the first member of the celluloid class of compounds.

This is considered the first man-made plastic.

Depending on how you wish to define "plastic" you can go even further back than that.

For example in 1844 Thomas Hancock in Britain patented the vulcanization of rubber (think car tires) and  Charles Goodyear patented it in the USA in the same year

In 1839 Eduard Simon, a German, discovered polystyrene

Around 1000 BC is the first historical evidence of the use of shellac. 

Source: https://en.wikipedia.org/wiki/Timeline_of_plastic_development

So plastic and its derivatives or components have been around for a long time (hmm wonder if any of those water bottles are still decomposing LOL) depending on how picky you want to be about the term

The first man made plastic was invented by  Alexander Parkes. He invented Parkesine in 1856.