Question

Identify the following as polynomial and hence write their degrees :

$x^2-\frac{1}{x^2}$

Answer 1

We have the algebraic expression :

$x^2-\frac{1}{x^2}$

Step 1 Identification of polynomial :

Since a polynomial is an algebraic expression in which powers of variables in each term are whole number.

Now $x^2-\frac{1}{x^2}=x^2-x^{-2}$ $\left(\because \frac{1}{a^m}=a^{-m},fromlawsofexponent\right)$

In term-1 that is $x^2$, power of variable is $2$ which is a whole number

In term-2 that is $x^{-2}$ , power of variable is $-2$ which is not a whole number

So power of all the variable in each term is not a whole number

Therefore, ${\mathit{x}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}$ is not a polynomial

Step 2 Finding degree of polynomial :

Since, $x^2-\frac{1}{x^2}$ is not a polynomial
Therefore, ${\mathit{x}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}$ does not have degree