Question

Which of the following is not a polynomial?
(This question has multiple correct options)
A.${x^2} + \dfrac{1}{x}$
B.$2{x^2} - 3\sqrt x + 1$
C.${x^3} - 3x + 1$
D.$2{x^{\dfrac{3}{2}}} - 5x$

Answer 1

Hint: Polynomial is an expression like ${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}}...{a_0} = 0$ having three conditions:
1.n must be a whole number
2.x cannot be in denominator
3.there are finite number of terms

Complete step-by-step answer:
In option A, ${x^2} + \dfrac{1}{x}$ , there is ‘x’ in the denominator which violates 2nd condition, so it is not a polynomial. Variables of a polynomial must be in the numerator having exponent as the whole number only.
In option B, $2{x^2} - 3\sqrt x + 1$ , $\sqrt x $ is not allowed in polynomials, the exponent must be a whole number, so this is also not a polynomial.
In option C, ${x^3} - 3x + 1$ , all three conditions are satisfied, as exponents are whole numbers, x is not in denominator and has three terms (finite), so this is a polynomial. The degree of polynomial is 3 which is the highest exponent. So, it is a cubic polynomial. It is also having three terms, so it is classified as a trinomial polynomial.
In option D, $2{x^{\dfrac{3}{2}}} - 5x$ , exponent is a rational number not whole number, so this is also not a polynomial.

Note: 1. In general polynomial, ${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}}...{a_0} = 0$ , ${a_0},{a_1},{a_2}....$ are called constants and ‘x’ is variable. Largest exponent of a polynomial is called degree of polynomial.
2.Constant polynomial is a polynomial which have degree 0
Zero polynomial is a polynomial having all constants i.e. ${a_0},{a_1},{a_2}....$ are equal to zero. The degree of zero polynomial is not defined.
3.There are also polynomials having more than one variable called multi-variable polynomials, which we will use in calculus in higher classes.