der Zusatz
additive
Something that is added to another thing in order to supplement, change, or improve it; an additive.
Dieses Lebensmittel enthält keine künstlichen Zusätze.
This food contains no artificial additives.

addition / additive
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additive
Something that is added to another thing in order to supplement, change, or improve it; an additive.
Dieses Lebensmittel enthält keine künstlichen Zusätze.
This food contains no artificial additives.

The word is derived from the verb 'zusetzen' and contains the prefix 'zu-', which often indicates a supplement or addition. A 'Zusatz' (addition/additive) is therefore something that is 'dazu gesetzt' (added to) something else.
Als Zusatz zu seinem Vortrag zeigte er noch ein kurzes Video.
As an addition to his talk, he showed a short video.
The context makes the meaning clear: If it is about food, drinks, or materials, it usually means an 'additive'. If it is about contracts, texts, or conversations, it means an 'addition' or 'supplement'.
Der Joghurt ist ohne den Zusatz von Zucker.
The yogurt is without added sugar.
In the meaning of 'additive', the word is often combined with adjectives such as 'künstlich' (artificial), 'chemisch' (chemical), or 'natürlich' (natural). On packaging, you often read 'ohne Zusatz von...' (without the addition of...).
Viele Fertigprodukte enthalten chemische Zusätze zur Konservierung.
Many ready-made products contain chemical additives for preservation.
When supplementary information is meant, the preposition 'zu' (+ dative) is often used to say what something is an addition to. For example, 'ein Zusatz zum Protokoll' (an addition to the minutes/record).
Er formulierte einen wichtigen Zusatz zum Gesetzestext.
He formulated an important addition to the legal text.
expansion / extension
'die Erweiterung' and 'der Zusatz' both have to do with something being added to something that already exists. 'Erweiterung' usually means that something becomes larger or more comprehensive overall, for example a building, a course, or a program. 'Zusatz' more often means a single additional element, a small piece of information, a remark, or also a substance that is added. The words can be confused because both describe adding something. When the overall scope grows, 'Erweiterung' is more natural. When just something extra is added, 'Zusatz' fits better.

Im Vertrag steht ein Zusatz zu den Kosten für die Reise.
There is an addition in the contract about the costs of the trip.
Frau Meier prüft die Erweiterung der Sicherheitsanlage zu Hause.
Ms. Meier is looking into expanding the home security system.