are the operator and promotor regions genes

Genes code for proteins. The regulatory elements like promoters and operators are not genes, nor are they transcribed into mRNA.

But then why does my book say that they are genes? Or maybe it's trying to say that in one aspect it's a gene as it's a length of DNA but we can't call it that as it doesn't code for polypeptides?

No idea. What book are you using? Each gene will have its own individual promoters, otherwise they won't get transcribed, so in that way you could say a promoter is part of a gene, but to say a promoter IS a gene is just wrong.

Operator and promoters are called regulatory sequences, because they regulate the activity of the gene. Proteins called Transcription Factors (TFs) will bind to the DNA at promotes sites and will stimulate transcription. They are not strictly speaking genes, rather regulatory sequences upstream of genes (Degree in Biomed) Just ask if you have more questions

What book are you looking at? if I read that somewhere I would disregard it as wrong tbh, but depends what book it is. The inducer I believe is a separate entity, coding for a distinct entity. It was a while since I did the lac operon, remember it confusing the hell out of me .