Domácí práce jsou důležité.

Because práce is one of those Czech nouns whose nominative singular and nominative plural look the same.

So in this sentence, you know it is plural because of the agreement:

• jsou = are
• důležité = plural adjective form

That tells you domácí práce is being treated as plural here.

Because it has to agree with práce in number and gender.

Here, práce is feminine plural, so the predicate adjective takes the plural form důležité.

A useful contrast:

• Ta práce je důležitá. = singular feminine
• Ty práce jsou důležité. = plural

So důležité is the correct plural form here.

Domácí is a soft adjective. In many basic forms, especially nominative singular and plural, it stays domácí regardless of gender and number.

So even though the noun is plural here, domácí still looks the same:

• domácí práce
• domácí úkol
• domácí zvíře

This is normal and very common in Czech.

It is in the nominative case, because it is the subject of the sentence.

In a sentence of the type X are Y, the subject is normally nominative:

• Domácí práce = subject
• jsou = verb
• důležité = predicate adjective agreeing with the subject

Because Czech does not use articles the way English does.

So Czech says simply:

• Domácí práce jsou důležité.

English needs something like Housework is important or Domestic chores are important, but Czech leaves definiteness to context.

Because Czech and English do not always package ideas the same way.

Czech often treats domácí práce as a set of separate tasks or chores, so plural agreement is natural:

• jsou důležité

English often uses the uncountable singular housework, even though the real-world meaning is similar.

So the grammar does not have to match word-for-word across the two languages.

Yes. Czech word order is much more flexible than English word order.

The neutral order here is:

• Domácí práce jsou důležité.

But you could also say:

• Důležité jsou domácí práce.

That version puts more emphasis on important. The basic meaning stays the same, but the focus changes.

Normally, no. In standard Czech, the present-tense forms of být are usually expressed in sentences like this.

So:

• Domácí práce jsou důležité. = normal full sentence

Without jsou, it would sound incomplete in ordinary speech, though you might see such omissions in headlines, notes, or very condensed styles.

Jsou is pronounced approximately like soh or s-yo blended very quickly, depending on how carefully someone is speaking.

A rough guide to the whole sentence:

• Domácí ≈ do-MAA-tsee
• práce ≈ PRAA-tseh
• jsou ≈ soh
• důležité ≈ doo-leh-zhi-tay

This is only an approximation for English speakers, but it can help you get started.

A singular version would be:

• Domácí práce je důležitá.

Here:

• je = singular is
• důležitá = singular feminine adjective

That is grammatically possible, but it gives a different grammatical reading from the plural sentence you have here. In your original sentence, Czech is clearly using the plural idea.