DawnC
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The only parent who has the right to claim a child as a dependent is the parent whom the dependent spent the most nights with.  That parent (the custodial parent) is entitled to the Child Tax Credit, and also to file Head of Household, claim the Earned Income Credit, and the Dependent Care Credit.  The other parent (the noncustodial parent) cannot claim the child.  

 

If the custodial parent is supposed to give (or wants to give) the dependent exemption to the noncustodial parent in certain years, that is done by the custodial parent giving the noncustodial parent a signed form 8332 Dependent Release.   This allows the noncustodial parent to claim the Child Tax Credit and exemption only.  The ability to qualify for Earned Income Credit, Head of Household, and the Dependent Care Credit, always stays with the custodial parent and can't be waived, shared, or transferred.

 

Where can you find 8332?   Please see Pub 501.

 

If the child spent more time with you, you claim the kid and get Head of Household but due to your income, do not get EIC or the child tax credit.   If the child spent more time with her, she would claim HoH, EIC and the child tax credit.   In either of these more common situations, no Form 8332 is needed.    If you qualify to claim the child, you could give her Form 8332, and she could take the child tax credit and only the child tax credit.  

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