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Hydrophilic means water liking or loving so it attracts H2O
This polymer is hydrophilic because of O and the H that makes it susceptible to be attracted to another water molecule easily because of its polarity.
Remember that like dissolves like. Water or H2O is polar, and polar compounds are likely to be dissolved in water (which is the definition of hydrophilic - water-liking).
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The H or hydrogens on this molecule do not make it hydrophilic because they are bonded to a carbon atom, they can not act as hydrogen bond doners because the the electronegative difference between hydrogen and carbon is so small it renders the bond non-polar thus not allowing it to participate in intermolecular hydrogen bonding.
The oxygen however can act as a hydrogen bond acceptor due to it having a high electronegativity in comparison to carbon and since there are oxygens every couple hydrogen this molecule can participate in many hydrogen bonds with water.
Water is very good at hydrogen bonding because it contains two (hydrogens) bond donators and one hydrogen bond acceptor (oxygen), thus it can form favorable interactions with this polymer allowing solubility of it.
Hope this is helpful