Thank you for your question. The Torah commands us to put tzitzit strings on the corners of our “garments”. There is a debate as to the which materials are considered “garments” according to the Torah, which materials were obligated in tzitzit only from a Rabbinical decree and which are exempt entirely from tzitzit.
The Shulchan Aruch, and the Serphadi practice, is to define a garment as one made from wool or linnen (see Shulchan Aruch, Orech Haim, 9,1). This is based on the opinion of Rav Nachman in Gemara Menachot 39b. The Rif (Tzitzit 14a) and Rambam Hilchot Tzitzit 3:2, and Sefer Hamitzvot Aseh 14 all rule in accordance with Rav Nachman. Other materials are considered as garments only Rabbinically. (Because of this, many Seprphadi Jews are careful to wear a woolen tzitzit garment in order to merit a Torah mitzvah, and not just a Rabbinic one). We will examine the status of polyester in a moment.
The Rema, (ibid), and the Ashkanazi practice, is to define a garment as one woven from any material. This is based on the opinion of Rava in Gemara Menachot 39b that all materials are able to be used to fulfill one's Torah obligation. Tosfot 39b s.v. VeRav cites the opinion of Rashi and Rabbeinu Tam in favor of Rava. According to this a cotton garment would also be obligated from the Torah in the mitzvah of tzitzit. None the less the Mishna Brurah (ibid 5) writes that one should wear a woolen garment in order to fulfill the mitzvah on a Torah level according to all opinions. Many Ashkanazi Jews follow this advice and wear a woolen tzitzit all day long – but others follow the practice of many great Rabbis (including the Gra and the Chazon Ish) who wore a cotton garment, and used wool only for their tallit which they said the blessing over in the morning. All opinions agree that a blessing may be said over non woolen tzitzit – because it is either obligated in the mitzvah according to the Torah, or from the Rabbis, and in either case needs a bracha.
Now we can approach the issue of a polyester garment. The Gemera states that a leather garment is exempt from tzitzit, and this is the halacha (see Shulchan Aruch, 10,4). Some Rabbis understand from this that synthetic materials are also exempt from tzitzit (see Igrot Moshe 2,1). Others though believe that if the synthetic thread is woven then it is obligated (at least Rabbinically) and only if it is manufactured as one sheet (like a sheet of plastic) would it be exempt (see Har Tzvi 1,9). Because of this argument some advise that one should refrain from saying a blessing over tzitzit on synthetic garments (Tzit Eliezer).
However, your question relates to a garment made from a cotton and polyester mix. The Mishna Brurah (9, 2 – and see also the Biur Halacha ibid s.v. Rechalim) rules that a garment made from two substances that are mixed together follows the majority. [This is true as long as the threads were made from a mixture – but if the warp is of one type of material and the woof of another it is not considered as following the majority.] Based on this if the majority of the garment is cotton, then all opinions would consider the garment as having the rule of a cotton garment.
So, in conclusion, a garment made of a majority of cotton with a minority of synthetic polyester, is definitely obligated in the mitzvah of tzitzit, and one should make a blessing over such tzitzit. According to the Sephardic tradition such a tzitzit is only obligated Rabbinically, and one may want to wear a woollen tzitzit (or at least a woollen Tallit) in order to merit fulfilling the mitzvah on a Torah level garment each day. According to the Ashkanazi tradition such a garment is obligated in tzitzit from the Torah, and making the effort to wear a woollen tzitzit (or tallit) is a stringency that many follow in order to merit to meet all opinions as to fulfilling the mitzvah from the Torah, and not just Rabbinically.