Question

If $x^g$+$\frac{1}{x^a}$ is a polynomial in one variable and if it is given that g = a, for what value of 'a' is this a polynomial?

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Answer 1

$x^g$+$\frac{1}{x^a}=x^g+x^{-a}$
Since g = a,
$x^g+\frac{1}{x^a}=x^a+x^{-a}$
For the expression $x^a+x^{-a}$ to be a polynomial, a and -a must be whole number so that the expression $\frac{1}{x^a}$ = 1 , hence a should be equal to 0 .
Thus, a = 0.

When a=0, the given polynomial will get simplified further as:

$x^g+\frac{1}{x^a}=x^a+x^{-a}$

$\Rightarrow$ $x^g+\frac{1}{x^a}=x^0+x^{-0}$

$\Rightarrow$ $x^g+\frac{1}{x^a}=x^0+x^0$

$\Rightarrow$ $x^g+\frac{1}{x^a}=1+1=2$

'2' is a constant polynomial in one variable with degree zero . Because $2=2x^0$.

So, 'a' must be equal to zero so that given expression is a polynomial in variable.