Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $ 4 x^{2}-3 x+7 $
(ii) $ y^{2}+\sqrt{2} $
(iii) $ 3 \sqrt{t}+t \sqrt{2} $
(iv) $ y+\frac{2}{y} $
(v) $ x^{10}+y^{3}+t^{50} $

To do:

We have to find which of the given polynomials is of one variable and which are not and state the reasons.

Solution:

(i) In $4x^2−3x+7$,

All the powers of $x$ are whole numbers.

Therefore, it is a polynomial in one variable $x$.

(ii) In $y^2+\sqrt2$

The power of $y$ is a whole number.

Therefore, it is a polynomial in one variable $y$.

(iii) $3\sqrt{t}+t\sqrt{2}$ can be written as,

$3\sqrt{t}+t\sqrt{2}=3t^{\frac{1}{2}}+ t\sqrt2$.

The exponent of the first term is $\frac{1}{2}$, it is not a whole number.

Therefore, it is not a polynomial.

(iv) In $y+\frac{2}{y}=y+2y^{-1}$.

The exponent of the second term is $-1$, which is not a whole number.

Therefore, it is not a polynomial.

(v) In $x^{10}+y^{3}+t^{50}$

It is not a polynomial in one variable as there are three variables $x$, $y$ and $t$.