Monomials and Polynomials

I am having a hard time grasping what makes something a monomial and then subsequently a polynomial. I get that division by a variable is not allowed and therefore not a monomial but why is that? I just like to make sense of these things as I find it much easier to retain if I understand something.

Also, why is something like 3^1/2x not a monomial when a square root of a constant by itself is? How is that any different than any other constant times a variable?

Thanks for the help.

Re: Monomials and Polynomials

"Mono" means "One", and "Rail" means "Rail"...

Seriously though, a monomial consists of ONE term, a polynomial consists of MANY terms (as "Poly" means "Many").

Re: Monomials and Polynomials

Quote:

Originally Posted by

**SilentEchoes** I am having a hard time grasping what makes something a monomial and then subsequently a polynomial. I get that division by a variable is not allowed and therefore not a monomial but why is that? I just like to make sense of these things as I find it much easier to retain if I understand something.

Also, why is something like 3^1/2x not a monomial when a square root of a constant by itself is? How is that any different than any other constant times a variable?

There is no universal agreement on this. See here.

However. there is general agreement that $\displaystyle \sqrt3~x$ is a monomial and that $\displaystyle \sqrt{3~x}$ is not a monomial.

Re: Monomials and Polynomials

Okay, well that is what I thought. Makes sense anyways, I guess my book is taking the 3^1/2x is not approach. Thanks for clarifying that it is not a universal agreement.

Again though, why is dividing by a variable not considered a monomial? What is the thinking behind this concept? (or is that what is not universally agreed upon?)

Re: Monomials and Polynomials

Because the terms "monomial" and "polynomial" are defined to refer exclusively to numbers multiplying the variables to non-negative integer powers.

Re: Monomials and Polynomials

Makes sense, thanks for the clarification.....but a constant by itself is considered a monomial. How does that fall into that definition? (sorry, just trying to understand)

Re: Monomials and Polynomials

Quote:

Originally Posted by

**SilentEchoes** Makes sense, thanks for the clarification.....but a constant by itself is considered a monomial. How does that fall into that definition? (sorry, just trying to understand)

__Zero__ is a non-negative integer.

Therefore, $\displaystyle \sqrt5~=~\sqrt5~x^0$ is a monomial.

Re: Monomials and Polynomials

Okay, I think I got it now. Appreciate the information.