For any empirical study to be considered an RCT, it needs to have one or more treatment groups that are compared to one or more control groups. Crucially, the allocation of subjects to the groups needs to be random. The "randomized" part of RCT in fact only refers to this allocation mechanism. Nothing else needs to be randomized.

Hence, any factorial experiment that complies with these specification is also a randomized controlled trial.

However, if, for example, the participants in a factorial experiment are not selected into a treatment group randomly, e.g. because whether someone receives a treatment or not is determined by an external non-random factor, such as gender, age, hair color, job, income, it is not an RCT.

Hope that helps!

Regarding the "Control Groups':

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The meaning of a control group is different in the factorial experiment.

In the TPRCT and the MACE there is typically one designated control group. By contrast, in a factorial experiment there is no single experimental condition that always constitutes the control group. Instead, each factor has its own control group, made up of a combination of conditions.

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Source:

'Factorial Experiments: Efficient Tools for Evaluation of Intervention Components'

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4171184/#!po=3.90625

For an RCT: See this link: https://www.bing.com/search?q=randomized+controlled+trial&qs=LS&pq=randomized&sk=LS1&sc=8-10&cvid=D922A82078F44E86A79F703E2F6F876D&FORM=QBRE&sp=2&ghc=1

For a factorial Design see: https://www.amazon.com/s?k=Douglas+montgomery+design+of+experiments&i=stripbooks&ref=nb_sb_noss

A fact. design may be part or all of an RCT. Best, David Booth