"قَلَم اَلْأُسْتاذة غالي."
Translation:The professor's pen is expensive.
قلم الأستاذة الغالي
Notice how we added AL to the adjective غالي, and by doing this we made it an attributive adjective following the compound of قلم الأستاذة (the professor's pen) - which is a Genitive relation between 2 nouns (and this compound is defined by having "AL" on the second noun in it).
Attributive adjectives follow the noun they describe in number, gender, and most importantly, definition. If you remove that from the adjective, it will be a predicative adjective telling an information about the professor's pen (i.e. the sentence in English would need to use the verb "to be" to connect the subject and the predicative).
The t sound comes from the ة. And the i sound after it from the genitive case. The genitive has endings -i/-in when the nominative has -u/-un.
So, with all vowels it is قَلَمُ الْأُسْتاذَةِ غَالٍي or transcribed qalam-u l2ustaadha-t-i ghaalii(-n).
Caveat: the final i-sound of غالي has nothing to do with the genitive but is just caused by the final ي.
None. Because you are mentioning the owner right away: الأستاذة
such relation is called Genitive, in which 2 nouns are added together and it resembles the English ('s) or (of) relationship. So, قلم الأستاذة (pen of professor) or (professor's pen). The possession here is controlled by the addition of (AL) to the second term in the compound الأستاذة (which you can imagine it as "of the" in English). The possessive articles or letters are added to the word only when the owner is not mentioned by name, and they are equal to the English (my, your, his, her,.... etc). We don't need those here because the Professor is mentioned indeed. Like, you wouldn't say in English: this is her the professor's pen - Same thing is applicable in Arabic.
By adding and removing the definite article (AL) from the predicate adjective.
- The professor's pen IS expensive: قلم الأستاذة غالي.
- The expensive professor's pen: قلم الأستاذة الغالي
In nominal sentences (sentences starting with a noun), the predicate of the sentence is typically indefinite. There are of course cases of the predicate being composite or a verb but those are special cases for now. The predicate of the nominal sentence must be indefinite.
In the second example, we have an attributive adjective and NOT a predicate. It is an adjective attached to the noun (or in this case the composite noun or genitive: قلم الأستاذة). The attributive adjective follows the noun it describes in everything: definition, gender, number. So, since the compound here is defined (قلم الأستاذة where the definite article is added to the second noun in the compound), the adjective غالي, being attributive, becomes الغالي.
You seem to be answering the wrong question. I'm asking about the difference between "the expensive professor's pen" (i.e. of the two pens in front of me, the pen of the professor who charges more for lectures) and "the professor's expensive pen" (i.e. of the professor's two pens, the pen that cost more money), and not about "the professor's pen is expensive".