Question

Which of the following is a polynomial?
A. $2x$
B. ${{x}^{2}}+{{y}^{-2}}-2{{z}^{2}}$
C. $5{{x}^{3}}{{y}^{2}}{{z}^{3}}$
D. $x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}$

Answer 1

Hint: Use the definition of polynomials to check whether the expression given in the four options are polynomial or not. Check for each of the four options one by one and then mark the options which are polynomial.

Complete step-by-step answer:
First we look at the definition of the polynomial.
A polynomial is an expression consisting of variables ( also called indeterminate) and coefficients, that involves only the operations of addition, subtraction, multiplication and non-negative integer exponents of variables.
If we look at options
A) $2x\Rightarrow $ it is an expression consisting of a single variable x and involves the operation multiplication and also the exponent of variable x is 1, which is a non-negative integer.
Therefore, 2x (option A) is a polynomial.
B) ${{x}^{2}}+{{y}^{-2}}-2{{z}^{2}}$
It is an expression consisting of three variables namely x, y and z and involves the operation addition and subtraction. The exponent of variable z and x is 2, which is a non-negative integer but exponent of variable y is (-2) which is a negative integer.
Hence ${{x}^{2}}+{{y}^{-2}}-2{{z}^{2}}$ (option B) is not a polynomial.
C) $5{{x}^{3}}{{y}^{2}}{{z}^{3}}$
It is an expression consisting of three variables namely x,y and z and it involves the operation multiplication only. the exponent of variable x.y and z is 3,2 and 3 respectively and these exponents of variables are non-negative integers.
Hence, $5{{x}^{3}}{{y}^{2}}{{z}^{3}}$ (option C) is a polynomial.
D) $x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}$
It is an expression consisting of a single variable x and it involves the operation of addition only. The exponent of the variable x are 1,2,3 and 4 and all these four exponents are non-negative integers.
Hence, $x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}$ (Option D) is a polynomial.
Therefore,2x, $5{{x}^{3}}{{y}^{2}}{{z}^{3}}$and $x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}$ are polynomial and ${{x}^{2}}+{{y}^{-2}}-2{{z}^{2}}$ is not a polynomial.
i.e. option A,C and D are polynomials.

Note: Students can easily solve this question by just checking for non-negative integral exponents of the variable before saying any expression is polynomial. In this question, all the given four options were simple, without involving any fraction, logarithms or exponential form. That is why, in this question, students just need to check the non-negative integral exponent of every variable present in the expression, because sometimes students make mistakes by checking one or two variables out of some more variables in the expression.