Answer :
Final answer:
Among the given functions, f(x) = 19x¹⁰-2x⁵+12x⁴-19x³+5x² and f(x) = x⁴ + 7 - 2x + 5x³ are polynomials because they fit the general criteria of having non-negative integer exponents for each variable. The first function is not a polynomial due to the division by a variable.
Explanation:
To determine which of the given functions are polynomials, it's crucial to understand the definition of a polynomial function. A polynomial function is an expression that can be written in the form a + bx + cx² + dx³ + ..., where each variable has a non-negative integer exponent, and a, b, c, d, etc. are constants.
- f(x) = 3x³ - (5)/(x³) is not a polynomial because it involves division by a variable.
- f(x) = 19x¹⁰-2x⁵+12x⁴-19x³+5x² is a polynomial because it fits the general form of a polynomial and all exponents are non-negative integers.
- f(x) = x⁴ - 4f(x) = 7 - 2x + 5x³ appears to be a typo but if we consider f(x) = x⁴ + 7 - 2x + 5x³, then it is a polynomial since all terms are in the correct form.
