Gra kooperacyjna jest gotowa.

Gra is a feminine singular noun in Polish. Because of that, any adjectives describing it must also be in the feminine singular form.

So in:

Gra kooperacyjna jest gotowa.

both adjectives match gra:

• kooperacyjna = feminine singular
• gotowa = feminine singular

If the noun were masculine or neuter, the adjective forms would change:

• Film kooperacyjny jest gotowy.
• Zadanie jest gotowe.

This matching of gender and number is very important in Polish.

Because kooperacyjna has to agree with gra, which is feminine.

Polish adjectives change depending on:

• gender
• number
• case

The basic dictionary form of many adjectives is often given in the masculine singular, so you might see kooperacyjny in a dictionary. But here, since gra is feminine singular, the correct form is kooperacyjna.

A quick comparison:

• kooperacyjny film = a cooperative game/movie? no, here film is masculine
• kooperacyjna gra = a cooperative game
• kooperacyjne zadanie = a cooperative task

For the same reason: gotowa must agree with gra.

The adjective gotowy means ready, but that is the masculine singular form. With a feminine noun like gra, it becomes gotowa.

Compare:

• On jest gotowy. = He is ready.
• Ona jest gotowa. = She is ready.
• Gra jest gotowa. = The game is ready.

So gotowa is simply the feminine form of gotowy.

Jest is the 3rd person singular form of the verb być = to be.

So:

• gra ... jest gotowa = the game ... is ready

In the present tense, Polish sometimes leaves out jest, especially in informal speech or in certain styles. So you may also see:

Gra kooperacyjna gotowa.

But the full, standard sentence is:

Gra kooperacyjna jest gotowa.

For learners, it is safest to use jest until you get comfortable with when omission sounds natural.

Yes, Polish word order is more flexible than English word order.

The most neutral version here is:

Gra kooperacyjna jest gotowa.

But other orders are possible, depending on emphasis:

• Kooperacyjna gra jest gotowa.
• Gotowa jest gra kooperacyjna.

These all express roughly the same basic meaning, but the emphasis changes:

• Gra kooperacyjna... focuses first on the game.
• Kooperacyjna gra... puts more attention on the type of game.
• Gotowa jest... emphasizes that it is ready.

For a learner, the original version is a very natural and safe choice.

The two adjectives do different jobs:

• kooperacyjna describes what kind of game it is: a cooperative game
• gotowa describes its current state: ready

So:

• gra kooperacyjna = cooperative game
• jest gotowa = is ready

In other words:

• one adjective is part of the noun phrase
• the other is linked by jest and works more like a predicate adjective

English does this too:

• The cooperative game is ready.

It is understandable, but in gaming contexts many speakers more often say kooperacyjny or use expressions based on kooperacja, and in everyday gamer language you may also encounter forms influenced by English, such as gra kooperacyjna or talk about co-op.

So gra kooperacyjna is understandable and acceptable, but depending on the exact context, other expressions may sound more idiomatic to some speakers.

If this is from learning material, the important grammar point is still the same:

• gra is feminine
• so kooperacyjna must be feminine too

Polish does not have articles like a, an, or the.

So gra can mean:

• a game
• the game

The exact meaning depends on context.

That means:

• Gra kooperacyjna jest gotowa. could mean
  • A cooperative game is ready
  • The cooperative game is ready

Usually the situation makes it clear which one is meant.

A rough English-friendly guide:

• gra ≈ grah
• kooperacyjna ≈ koh-oh-peh-ra-TSYI-na
• jest ≈ yest
• gotowa ≈ goh-TOH-va

A few useful pronunciation notes:

• g is always hard, like in go
• j sounds like English y
• w sounds like English v
• y is a vowel that does not exist exactly in English; it is not the same as ee
• cz / sz / ś / ć are often tricky in Polish, but this sentence avoids most of those

The stress in Polish is usually on the second-to-last syllable:

• kooperacyjna
• gotowa

It is in the nominative case.

Why? Because it is the subject of the sentence:

• Gra kooperacyjna = the thing that is ready

In dictionary form, nouns are usually given in the nominative singular, and gra is exactly that form.

The adjective kooperacyjna also appears in the nominative feminine singular to match it.

Yes. That means This cooperative game is ready.

Adding ta makes the noun phrase more specific:

• gra kooperacyjna jest gotowa = the/a cooperative game is ready
• ta gra kooperacyjna jest gotowa = this cooperative game is ready

Again, ta must agree with gra, so it is feminine singular:

• ten for many masculine nouns
• ta for feminine nouns
• to for neuter nouns

Usually gotowa means ready, not necessarily finished.

So Gra kooperacyjna jest gotowa most naturally means something like:

• the cooperative game is ready to be played
• the cooperative game is prepared
• the cooperative game is complete and available

Whether it means prepared or finished depends on context.

If you specifically want to say a game is finished/completed, Polish might use other wording depending on what exactly you mean, for example whether development is finished, a match is over, or something has been prepared.

The plural would be:

Gry kooperacyjne są gotowe.
= Cooperative games are ready.

Changes:

• gra → gry
• kooperacyjna → kooperacyjne
• jest → są
• gotowa → gotowe

This shows agreement again:

• plural noun
• plural adjectives
• plural verb

So Polish keeps all the parts aligned with each other.