Question

Which of the following is not a polynomial?
(a) 3
(b) $3{{x}^{2}}+\sqrt{5}x$
(c) ${{x}^{2}}+\dfrac{1}{{{x}^{2}}}-4$
(d) $\pi {{t}^{7}}-3{{t}^{2}}+4$

Answer 1

Hint: Understand the definition and examples of polynomials. Check each option one by one and eliminate the options which have a polynomial expression. Pick the odd one out to get the answer.
Here, we have been provided with four options containing different expressions. We have to determine the option which does not contain a polynomial expression. To do this first, we need to know what is a polynomial.

Complete step-by-step solution:
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Examples of polynomials can be, ${{x}^{2}}-4x+8$ and $x{{y}^{3}}+y{{z}^{2}}+1$. The first polynomial contains one variable x and the second polynomial contains three variables, x, y, and z.
On reading the above paragraph, we see that a polynomial involves non-negative integer exponentiation of variables. This means that the exponent or power of the variable must not be negative. It means that the exponent can be 0 or positive only. Also, note that it should be an integer.
Now, let us check the options one by one.
(a) 3
This can be written as $3{{x}^{0}}$. Since ${{x}^{0}}$ can be a polynomial term, therefore 3 is a polynomial.
(b) $3{{x}^{2}}+\sqrt{5}x$
Here we can see that the exponents of x are 1 and 2. Therefore, $3{{x}^{2}}+\sqrt{5}x$ is a polynomial.
(c) ${{x}^{2}}+\dfrac{1}{{{x}^{2}}}-4$
The above expression can be written as ${{x}^{2}}+{{x}^{-2}}-4$. Clearly, we can see that the power of x is 2 and -2. Since there is a negative element, so this expression is not a polynomial.
(d) $\pi {{t}^{7}}-3{{t}^{2}}+4$
Here, the variable is ‘t’ in place of ‘x’, but the process will remain the same. Exponents of ‘t’ are 7 and 2. Therefore, $\pi {{t}^{7}}-3{{t}^{2}}+4$ is a polynomial.
By checking all the options, we can say that ${{x}^{2}}+\dfrac{1}{{{x}^{2}}}-4$ is not a polynomial.
Hence, the option (c) is our answer.

Note: One may note that option (a) is a polynomial. It is a constant polynomial because the exponent of ‘x’ is zero here. Here, we must check all the options because we are finding an odd one out. That means we are eliminating the options one by one. Finally, remember that we must know the definition of the polynomial to solve this problem.