Is it possible for a 2nd-degree monomial to have 3 variables? (What about $6p^0qr$?)

Question

Is it possible for a 2nd-degree monomial to have 3 variables?

I think it's not, and then I remembered this: $6p^0qr$. But the $p^0$ is $1$, so for me it's not an example of a 2nd-degree monomial with three variables.

What do you think?

Answer 1

A second degree monomial takes on the form of:

f(x) = ax^2

If you want the monomial to take on three variables then you could have:

f(x,y,z) = ax^2

Technically this would still be a function of three variables. Or you could have:

f(x,y,z) = a(xyz)^2

But I don't think this would be considered a polynomial anymore.

In your own example you have:

6p^oqr

Here p would be the variable and the oqr would have to be constants for the monomial to be degree 2. In other words:

oqr=2

Answer 2

A monomial in 3 variables (say $x,y,z$) is of the form $cx^py^qz^r$ where $c$ is a nonzero constant and $p,q,r$ are positive integers (so each at least $1$).

The degree of the monomial is $p+q+r$, so it must be at least $3$. This makes degree $2$ impossible.

Note: this assumes the minimal variable list contains three variables; that is, all three variables contribute to the value of the monomial. In other words, there are strictly three variables. But this is no more problematic than stating that $x^2+0$ is not a binomial.