You cannot get education credits when you file married filing separately. That is one of the many disadvantages of filing MFS instead of filing a joint return. Sorry.
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Although there is no one answer since every situation is different, generally filing jointly will give you a bigger refund or less taxes due. When you file separately, your tax rate is higher and you won't be able to claim:
Itemized deductions if your spouse has already claimed the standard deduction, or the other way around.
On top of that, if you live in the community property states of Arizona, California, Idaho, Louisiana, Nevada, New Mexico, Texas, Washington, or Wisconsin, you have to deal with community property allocations and adjustments, which adds extra work and complexity to your tax preparation chore.