Which of the following expressions are not polynomials?:
(i) $x^2+2x^{-2}$
(ii) $\sqrt{ax}+x^2-x^3$
(iii) $3y^3-\sqrt{5}y+9$
(iv) $ax^{\frac{1}{2}}y^7+ax+9x^2+4$
(v) $3x^{-3}+2x^{-1}+4x+5$

Given:

The given expressions are:

(i) $x^2+2x^{-2}$

(ii) $\sqrt{ax}+x^2-x^3$

(iii) $3y^3-\sqrt{5}y+9$

(iv) $ax^{\frac{1}{2}}y^7+ax+9x^2+4$

(v) $3x^{-3}+2x^{-1}+4x+5$

To do:

We have to find which of the given expressions are polynomials.

Solution:

Polynomials: 

Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.

To identify whether the given expression is polynomial, check if all the powers of the variables are whole numbers after simplification. If any of the powers is a fraction or negative integer then it is not a polynomial.

(i) The given expression is $x^2+2x^{-2}$.

The term $2x^{-2}$ has a negative power of $-2$.

Therefore,

The given expression is not a polynomial.

(ii) The given expression is $\sqrt{ax}+x^2-x^3$.

The term $\sqrt{ax}=\sqrt{a}x^{\frac{1}{2}}$ has a fraction power of $\frac{1}{2}$.

Therefore,

The given expression is not a polynomial.

(iii) The given expression is $3y^3-\sqrt{5}y+9$.

The given expression does not have any negative or fractional powers.

Therefore,

The given expression is a polynomial.

(iv) The given expression is $ax^{\frac{1}{2}}y^7+ax+9x^2+4$

The term $ax^{\frac{1}{2}}y^7$ has a fraction power of $\frac{1}{2}$.

Therefore,

The given expression is not a polynomial.

(v) The given expression is $3x^{-3}+2x^{-1}+4x+5$.

The terms $3x^{-3}$ and $2x^{-1}$ have negative powers of $-2$ and $-1$.

Therefore,

The given expression is not a polynomial.

Updated on: 13-Apr-2023

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