Option A,C and D are not polynomials.
Step-by-step explanation:
Polynomial functions are given by
p(x) = a₀ + a₁x¹+ a₂x²+ ...........+aₙxⁿ
Where a₀, a₁, a₂, ..., an are constant coefficients and n is non negative integer.
Option A
[tex]p(x)=\frac{2}{x^2}+x+1\\\\p(x)=2x^{-2}+x+1[/tex]
Here one exponent of x is -2, so this is not a polynomial function.
Option B
[tex]p(x)=x^2+\sqrt{2}x+1[/tex]
Here all the exponents are non negative integer, so this is a polynomial.
Option C
[tex]p(x)=x^2+2\sqrt{x}+1[/tex]
Here one exponent of x is 0.5, so this is not a polynomial function.
Option D
p(x)= x⁻²+x+1
Here one exponent of x is -2, so this is not a polynomial function.
Option E
[tex]p(x)=\frac{2}{3}x^2+x+1[/tex]
Here all the exponents are non negative integer, so this is a polynomial.
Option A,C and D are not polynomials.
