In Canada, pesticides are regulated: they must be registered to be used. Should one wants to register a mixture of synthetic pesticide and plant extract, he has to go through the same process.
Worldwide, synthetic pesticide use needs to be registered through a central regulatory body in each jurisdiction, as indicated by Charles. In many countries, plant derived insecticides also require registration through the same or a similar process unless, as in the USA, some chemicals of natural origin already have an exemption for use within certain limitations (either concentration or usage).
However, mixtures of synthetic and plant derived pesticides should have a further evaluation for registrations because, despite a widespread misunderstanding and incorrect belief, plant derived pesticides are not harmless, or in many cases not even "safe" for widespread use. Consequently, a mixture of a synthetic pesticide and a natural pesticide will be to some extent, possibly a great extent, more toxic than the sum of the individual toxicities of the different chemical entities.
Therefore, the regulatory process should be at least as rigorous for a mixture and possibly more rigorous than for individual compounds. Unfortunately, many regulatory bodies have not yet fully comprehended this trend for mixed chemical use or the safety implications of using them.
The basic idea for making combinations of chemical pesticide with the plant based ones is to minimize the dose of chemical pesticide and overall enhancement of activity against a test organism. I do agree with above statement that at times these combinations may pose even undesirable toxicities which need further comprehensive studies for risk assessments against non target organism in the same ecosystem and mammalian cell lines as well. However, laboratory trials of preregistered chemical pesticides which are reported in literature may be taken as per my understanding.
Of course all the answers above a scientifically sound. Plant extracts are not entirely harmless depending on the plant material from which they were obtained. However, they may be less persistent in the ecosystem especially in tropical settings. I do not think that mixing both synthetic and botanical pesticides is a great idea, you should consider applying them in sequence with the botanical first, if the results aren't as great follow up with a low-dose narrow spectrum synthetic pesticide