A polynomial is a sum of finite number terms with real coefficients and non negative or positive integral powers of the variable.
The polynomial can not have fractional power of the variable. A polynomial has no terms with negative exponents of the variable.
A polynomial can have one term. It can have two or more terms also.
A polynomial can be with single variable or many variables. If the highest of the term and the degree of the polynomial are the same.
c is not a polynomial in the given set, as it conains a term -4/t whose degree is -1 as -4/t = -4t^(-1). The exponent of the variable t of this term is -1 , a negative exponent. A polynomial does not have a term with a negative exponent variable.
A polynomial function is of the form that can be represented as:
f(x) = A1*x^n + A2*x^(n-1)....An*x^0
Now looking at the three functions given:
a. y = 3x^2 + 2x + 1/2 = 3x^2 + 2x + 0.5
This is a polynomial as it fits in the format given exactly.
b. y = 10X + 8
This also is a polynomial as it also fits into the format given.
c. y= t^2 + 2t - 4/ 2t-1
Here y = t^2 + 2t - 4/ (2t-1)
This cannot be expressed as the sum of a series of terms with a decreasing power of t due to the term (2t-1) which lies in the denominator.
Hence the option c is not a polynomial.