Question

How to solve this showing all work ! Also explain why they aren’t polynomials if you can :)

Answer 1

Polynomials are linear combinations of positive integer powers. For example,

$x^{100}+30x^2-\sqrt 2\,x+21$

is a polynomial, but

$-x^{-1}-4x^\pi+x^{3/2}$

is not.

(In case you're unfamiliar with the term, a linear combination of two quantities is the sum of scaled versions of those quantities, e.g. a general linear combination of $a$ and $b$ is $k_1a+k_2b$, where $k_1,k_2$ are any numbers.)

$8x^2-18\sqrt x+5x-3$ is not a polynomial; the offending term is $-18\sqrt x$, or $-18x^{1/2}$.