In my opinion when someone writes 3x^2=0 **they are not thinking** of 3x^2 as the product of two factors, namely 3 and x^2. If a product of factors equal 0, I feel that you should set each factor to 0 and solve. Especially when many students claim that 3=0 when x is -3

They

**might **be thinking that way. But there's no evidence that anyone here did, and you can't read minds any better than I can.

And in saying "3x^2 [is] the product of two factors, namely 3 and x^2", are you denying that x is a factor? Of course not.

Yes, your way of saying it (which I sometimes do, too, when I first introduce the topic) is a way to emphasize that we are applying the "zero-factor theorem". However, I do that not because students

**don't** see 3 as a factor, but because

**a few do**, and get confused, thinking that 3 = 0 means x = 3. So I'm preemptively keeping them from going too far in that direction, and then tell them that they can ignore constant factors.

So a student should quickly move on from there and realize that they can just write 3x^2=0 and know that x = 0. So in teaching, go ahead and say what you did; but don't imply that it's wrong to say it in other ways.

One of the benefits of a forum like this is that different helpers may diagnose a student's problem in different ways, and something I don't think of, that you do, may be just what they need. Or it may not. In any case, it's not something to argue over. We're here to join forces in helping students, not to quibble over different styles.