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PHB page 168 says:

Polearm Master: [...] When you take the Attack action and attack with only a glaive, halberd, or quarterstaff, you can use a bonus action to make a melee attack with the opposite end of the weapon. This attack uses the same ability modifier as the primary attack. The weapon’s damage die for this attack is a d4, and it deals bludgeoning damage.

Does this bonus attack benefit from ability modifiers for damage?

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    \$\begingroup\$ I can't believe we did not have this question before. \$\endgroup\$ – András Mar 4 at 7:51
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Yes.

All weapon damage rolls add their corresponding modifier to the damage unless a feature tells you otherwise. This is from the rules on Damage Rolls in the Combat chapter of the Basic Rules:

When attacking with a weapon, you add your ability modifier--the same modifier used for the attack roll--to the damage.

Since you're making an attack with the weapon, you add the modifier. It doesn't matter that you're making that attack because of a feat, or with the opposite end of a weapon, or using a bonus action, or so on. Nothing in the feat suggests an exception to the general rule, so add that modifier.

Note that there are situations where the rules allow you to make an attack with a weapon but deny you a modifier to the damage, such as when using two-weapon fighting ("don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative"), but those are exceptions to the general rule.

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Yes

The feat says you

use the same ability modifier as the primary attack.

Normally, if you use Strength to attack, then you add your Strength modifier to the damage. But it could potentially be a different ability (e.g. a Druid that has cast Shillelagh would use Wisdom). This is a basic rule of combat damage adjudication.

The rules on Damage Rolls state (Basic Rules, p. 77):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. [...] When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll—to the damage.

This has already come up in a tweet by Jeremy Crawford:

"does a player using the polearm master feat add strength to the damage roll of the bonus attack?"

Yep!

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    \$\begingroup\$ Do note that tweets are no longer an official source for rulings so it would help if you back this up with anything from the book as well. \$\endgroup\$ – Rubiksmoose Mar 4 at 12:36
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    \$\begingroup\$ You should also quote the relevant content from the tweet if you use it as support for your answer. As for the rules basis for the ruling, only the TWF rules specify you don't add ability score mod; every other "bonus-action attack" granted by a feature or feat that I know makes no such exception, so you add your ability score mod as normal for any attack. \$\endgroup\$ – V2Blast Mar 4 at 13:41
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    \$\begingroup\$ Edit was to put your bottom line up front, and then support from dev. I think that organization fits better, revert if you find it does not. I threw in the basic rules citation on damage as a bonus. :) \$\endgroup\$ – KorvinStarmast Mar 4 at 14:27

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