$\text{All the variables of a polynomial}\text{should have whole numbers as}\text{exponents.}$
$\text{For}x+x^2,\text{the exponents of the}\text{variable x are 1 and 2. All the}\text{exponents are whole numbers.}\text{Therefore, it is a polynomial.}$
$\text{In the expression}\sqrt[3]{3y}+y^2,\text{exponent}\text{of}\sqrt[3]{3y}\text{is}\frac{1}{3},\text{but}\frac{1}{3}\text{is not a whole}\text{number.}\text{Hence,}\sqrt[3]{3y}+y^2\text{is not a polynomial.}$
$\text{For}x+\sqrt{3},\text{the exponent of the}\text{variable is 1 which is a whole number.}\text{Therefore, it is a polynomial.}$
$\text{For x, the exponent of the variable is}\text{1 which is a whole number.}\text{Therefore, it is a polynomial.}$