Question

Which of the following expressions are polynomials in one variable and which are not?

$\left(i\right)4x^2-3x+7$

$\left(ii\right)y^2+\sqrt{2}$

$\left(iii\right)3\sqrt{t}+t\sqrt{2}$

$\left(iv\right)y+\frac{2}{y}$

$\left(v\right)x^{10}+y^3+t^{50}$

Answer 1

$\left(i\right)4x^2-3x+7$
The given expression is $4x^2-3x+7$.

The exponents of the variable in the expression are 2, 1 and 0, which are whole numbers.

Also, there is only one variable, x, in the expression.

Since the expression contains only one variable, and the exponents of the variable are whole numbers, it is a polynomial in one variable.

$\left(ii\right)y^2+\sqrt{2}$
The given expression is $y^2+\sqrt{2}$.

The exponent of the variable in the expression is 2, which is a whole number. Also, the expression contains only one variable, y.

Therefore, the above expression is a polynomial in one variable.

$\left(iii\right)3\sqrt{t}+t\sqrt{2}$
The given expression is $3\sqrt{t}+t\sqrt{2}$.

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

Here, the exponents of the variable are $\frac{1}{2}$ and $1$.
As $\frac{1}{2}$ is not a whole number, the expression is not a polynomial.

The expression has only one variable, t.

Hence, it is an expression of one variable but not a polynomial.

$\left(iv\right)y+\frac{2}{y}$
The given expression is $y+\frac{2}{y}$.

$\Rightarrow y+\frac{2}{y}=y+2y^{-1}$.

The above expression has –1 as an exponent, which is not a whole number. Therefore, it is not a polynomial.

The expression contains only one variable, y.

Hence, it is an expression of one variable but not a polynomial.

$\left(v\right)x^{10}+y^3+t^{50}$
The given expression is $x^{10}+y^3+t^{50}$.

The exponents of the expression are 10, 3 and 50.
As all the powers are whole numbers, it is a polynomial.

Also, the expression possesses three variables, x, y and t.

Therefore, the expression is a polynomial but not a polynomial in one variable.