Question

Which of the following is not a polynomial?
(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: For solving this problem, we consider all the respective cases one by one. To analyse whether the given equation is a polynomial, we just see the exponent of x whether it is a positive integer or not. If it is negative or fractional, then the considered case is not a polynomial.

Complete step-by-step solution -

In mathematics, a polynomial is defined as an expression having a combination of variables and coefficients. These are interlinked with each other by mathematical operators such as addition subtraction multiplication.
According to our problem, we have to identify the invalid polynomials from the four given cases. An invalid polynomial is the one having negative integer or rational exponent of variables.
For case 1, the second term has a negative exponent so it is not a polynomial. $\left( {{x}^{2}}+\dfrac{1}{x} \right)$
For case 2, the second term has fractional exponents so it is also not a polynomial. $\left( 2{{x}^{2}}-3\sqrt{x}+1 \right)$
Case 3 is a polynomial because it has positive integral exponent. $\left( {{x}^{3}}-3x+1 \right)$
For case 4, the first term has a fractional exponent so it is also not a polynomial. $\left( 2{{x}^{\dfrac{3}{2}}}-5x \right)$
Therefore, option (a), (b) and (d) are correct
.
Note: The key concept involved in solving the problem is the knowledge of verifying the validity of polynomials. Alternatively, by remembering the very basic definition of polynomials, students can tick mark the correct answers by observation.