Я забыл шарф дома, поэтому мне холодно.

I forgot my scarf at home, so I am cold.

Breakdown of Я забыл шарф дома, поэтому мне холодно.

• я: I
• дома: at home
• мне: me
• забыть: to forget

Questions & Answers about Я забыл шарф дома, поэтому мне холодно.

Why is it забыл and not забывал?

Забыл is past tense of the perfective verb забыть and refers to a single completed event: you ended up not having the scarf because you forgot/left it behind.
Забывал (imperfective, забывать) would usually mean something like used to forget (repeatedly) or would focus on the process/background rather than the completed result.

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Does Я забыл шарф дома mean “I forgot the scarf at home” or “I forgot about the scarf at home”?

In this structure, it means you left it at home / forgot to take it: забыть + object + location → забыть шарф дома = to forget/leave the scarf at home.
If you want “forgot about the scarf,” Russian typically uses забыть о + prepositional: Я забыл о шарфе.

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Why is шарф in this form (no ending change)?

Because шарф is a masculine noun and here it’s the direct object of забыл, so it’s in the accusative case. For many masculine inanimate nouns, the accusative equals the nominative: шарф → шарф.

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Why is it дома and not в доме?