It's essential you understand precisely what "polynomial" means.
Characteristics of polynomials:
1) All powers of x are zero or positive integers. This eliminates any powers of x which show up in the denominator of any term. No square or cube roots, for example.
2) Coefficients are real numbers, but need not be integer.
3) Generally you'll have a function name (such as "y") on the left of the " = " sign and ONLY powers of the independent variable (such as "x") on the right side.
In the list given, I see 2 polynomials. In 6 + w appears the first power of w; that integer power is 1. In the second I see y with the fractional exponent (1/3); this expression is NOT a polynomial. In the third, we have the same situation as in the first (6+w); this is a poly. In the fourth and last expression, I see a mixture of x and y. y = 2x^4 would be poly, but 2x^4 - y would not.