Which of the following are polynomials? In case of polynomial, write its degree:
$x^{5} - \frac{3}{x} + \frac{1}{3} x^{2} - 4$

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Solution

Definition:
An expression of the form $a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . + a_{n}$ where $a_{0}, a_{1}, a_{2}, . . ., a_{n}$ are real numbers and n is a whole number, is called a polynomial.
The highest power of the variable in a polynomial is called its degree.

$x^{5} - \frac{3}{x} + \frac{1}{3} x^{2} - 4 = x^{5} + \frac{1}{3} x^{2} - 3 x^{- 1} - 4$
Given expression $x^{5} - \frac{3}{x} + \frac{1}{3} x^{2} - 4$ is not in the form of $a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . + a_{n}$ where $a_{0}, a_{1}, a_{2}, . . ., a_{n}$ are real numbers and n is a whole number.
Because, the exponent of x is not a whole number.
Thus, it is not a polynomial.

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