"Fully support" is not enough. The child must live with you.
There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.
A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:
He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
He lived with the parent (including temporary absences such as away at school) for more than half the year
You follow the TurboTax (TT) interview. TT will confirm that you can claim the dependent and give you the $500 other dependent (over age 17) credit. To claim the tuition credit, enter education info at: