Question

Which of the following expression(s) is/are not representing a polynomial for $n\in N$.

• A. $x^n+x^{\left(\frac{1}{n}\right)}+\frac{1}{n}$
• B. $x^{n!}+x^{\left(\frac{n}{n!}\right)}+\frac{1}{n!}$
• C. $x^{n!}+x^{\left(\frac{n!}{n}\right)}+\frac{1}{n!}$
• D. $x^{\left(\frac{n^2-1}{n+1}\right)}+x^2$

Answer 1

Answer: (A), (B)

$x^{n!}+x^{\left(\frac{n}{n!}\right)}+\frac{1}{n!}$

An algebric expression containing one,two or more dissimilar terms with non-zero coefficients (with variable having non-negative integers as exponent) is called a polynomial.

In the given options: the exponents, $n!,\frac{n!}{n}=(n-1)!$ and $\frac{n^2-1}{n+1}=n-1$ are non-negative number $\forall n\in N$, and the exponent $\frac{1}{n},\frac{n}{n!}=\frac{1}{(n-1)!}$ are not integers for all values of $n\in N$.
So, options $A$ and $B$ do not represent a polynomial.