How come x^4/3 is a ploynomial, but 3/x^4 isn't?
this site gives a good explanation.
http://www.purplemath.com/modules/polydefs.htm
basically, in a regular polynomial the exponents must be whole numbers
since 3/x^4 = 3x^-4 , this violates that definition
x cannot have a negative power in a polynomial by definition.
http://www.merriam-webster.com/dictionary/polynomial
A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division or square roots. The exponents of the variables must be non-negative integers.
In the case of x^(4/3), it can be written as x^(4*(1/3)), where 4 is a non-negative integer and 1/3 is a rational number. This rational exponent can also be expressed as a cube root (∛) of x^4. Since the exponent is a non-negative rational number and not involving division or square roots, it can be considered a polynomial.
However, the expression 3/x^4 involves division, where x^4 is in the denominator. Although the base x^4 is a non-negative integer exponent, the presence of division makes it not a polynomial. Division is an operation that is generally not allowed in polynomials since it introduces fractional or negative powers.
In summary, x^(4/3) is a polynomial because it satisfies the criteria of having non-negative integer exponents and not involving division or square roots, while 3/x^4 is not a polynomial because it involves division.