Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
$( i).\ 4x^{2}-3x+7$
$( ii).\ y^{2}+\sqrt{2}$
$( iii).\ 3\sqrt{t}+t\sqrt{2}$
$( iv).\ x^{10}+y^{3}+t^{10}$
$( v).\ y+\frac{2}{y}$
To do: To find which of the polynomials are of one variable and which are not and to state the reason also.
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 $y^2+2$
The power of $y$ is a whole number.
Therefore, it is a polynomial in one variable $y$.
$( iii)$. In $3\sqrt{t}+t^2=3t^{\frac{1}{2}}+ t^{2}$.
The exponent of first term is $\frac{1}{2}$, it is not whole number.
Therefore, it is not a polynomial.
$( iv)$. In $x^{10}+y^{3}+t^{10}$
It is not a polynomial in one variable as three variable $x$, $y$ and $t$ occur in it.
$( v)$. In $y+\frac{2}{y}=y+2y^{-1}$.
The exponent of first term is $−1$, which is not whole number.
Therefore, is it not a polynomial.
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