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Identify which of the following are polynomials and which are not.

The expression $-2x^9-\frac{7}{13}x^3-1$ a polynomial.

The expression $5x^{-2}-5x^2+3$ a polynomial.

The expression $\sqrt{2}x-\frac{3}{5}$ a polynomial.

The expression $5x^3-5^{-5}x-x^4$ a polynomial.

The expression $\frac{25}{x^2}+23-x$ a polynomial.

The expression $37x^6-x+8^{\frac{4}{3}}$ a polynomial.

The expression $\sqrt{7x}-4x^3$ a polynomial.

The expression $6x^{\frac{3}{2}}+1$ a polynomial.

The expression $6^x-3x^6$ a polynomial.

Solution

The expression $-2x^9-\frac{7}{13}x^3-1$ is a polynomial.

The expression $5x^{-2}-5x^2+3$ is not a polynomial because it has negative exponents on a variable.

The expression $\sqrt{2}x-\frac{3}{5}$ is a polynomial. Note that coefficients can have radicals even though variables cannot, and the square root here is only applied to the $2\text{.}$

The expression $5x^3-5^{-5}x-x^4$ is a polynomial. Note that coefficients can have negative exponents even though variables cannot.

The expression $\frac{25}{x^2}+23-x$ is not a polynomial because it has a variable in a denominator.

The expression $37x^6-x+8^{\frac{4}{3}}$ is a polynomial. Note that coefficients can have fractional exponents even though variables cannot.

The expression $\sqrt{7x}-4x^3$ is not a polynomial because it has a variable inside a radical.

The expression $6x^{\frac{3}{2}}+1$ is not a polynomial because a variable has a fractional exponent.

The expression $6^x-3x^6$ is not a polynomial because it has a variable in an exponent.