I have this very trivial question, but I think I might have interpreted it incorrectly.
The expression shown below is a polynomial of what degree? $$x^3 {(x+\frac{1} {x})} {(1 + \frac {1} {x+1} + \frac {1} {x^4})}$$
I was going to say degree 4, but this 'polynomial' technically not a polynomial as it will have negative exponents. What is correct here?
The above is not a polynomial - polynomials are strictly meant to have non-negative integers as their exponents, ie. for all exponents in a polynomial expression $k$, $k∈ℤ^+$.
Polynomyals cannot have rational functions in it's expression. This is becaulse in general polynomyals are defined as functions wich can be generated by multiplication and addition of corresponding ring and is generated from one variable. So to construct polynomyal we can use just multiplication and addition. Substraction is just multyplyig $x$ by $-1$. So division is not allowed. In that definition this expression isn't polynomyal and isn't possible to find it's degree. But if you really need such thing to find read about Laurent series wich allows to express function on complex domain into sum of linear combinations of element of $\mathbb{C}$ and powers of $x$ including negative ones.
