Is this a polynomial ( true or false ) ?

First of all a binomial is a special polynomial just as a trinomial is a polynomial.  

Since we're talking about definitions, let's be precise, although this might seem petty and pedantic.

First of all, it's an equation, not an expression, so right off the bat it's false. Even "f(x) = x^3 + 2x" is not a polynomial. That would be a polynomial function.

To be a polynomial, an expression has to have integer exponents >= 0 (the constant term would be cx^0).

So x^3 + 2x^1/3 is not a polynomial.

So it's a function, but not a polynomial function.

Also, your reasoning is incorrect.

x^4 + 7x is a polynomial and a binomial. An expression can be both a polynomial and a binomial.

g(x) = x^4 + 7x is a polynomial function.

If you mean:

f(x) = x³ + 2x^(1/3)

Then it isn't a polynomial because the exponent isn't an integer.

If it was:

f(x) = x³ + (2/3)x

Then yes, that would be considered a polynomial.

You need parentheses if you want ⅓ to be the exponent. As written you have:

f(x) = x^3 + 2x^1/3

= x^3 + 2x/3 

= x^3 + (2/3)x

Anyway, I believe you meant:

f(x) = x^3 + 2x^(1/3)

The issue is not the number of terms (a binomial or even a monomial is still a polynomial).

The reason it is not a polynomial is that all of the terms must have a whole number exponent (non-negative integer).

Because the exponent is a fraction (1/3), this is not a polynomial.

Answer:

False

Read more about polynomials at the link below.