Learn Polish: Nie znam tylu słów po polsku. - I don't know that many words in Polish.

The verb znam is negated (nie znam), which triggers the genitive case, often called the Genitive of Negation. While tyle is the standard nominative and accusative form for non-masculine-personal nouns like words, its genitive form is tylu.

If the sentence were positive, would it still be tylu słów?

No, if it were positive, it would be Znam tyle słów. An affirmative znam takes the accusative case, so you use tyle. The noun słów remains in the genitive plural in both versions because quantifiers like tyle always force the noun that follows them into the genitive plural anyway.

How does słowo become słów in the genitive plural?

For most neuter nouns ending in -o, you form the genitive plural by simply dropping the final -o. When dropping that vowel leaves a word ending in a consonant, the preceding 'o' often changes to 'ó' (o with an acute accent). This gives us słów.

Could I use tak dużo instead of tylu?