Which of the following is a polynomial? Select all that apply. \( f(x)=3 x\left(x^{4}-6 x^{3}+1\right) \) \( g(x)=5 x^{7}+2 x^{-3}-7 x+10 \) \( y=8 \sqrt{x}+6 x-4 \) \( h(x)=5^{x}+8 x^{3}-2 x \) \( f(x)=-3 x^{4}+\frac{5}{2} x \) \( f \)

To determine which of the given functions are polynomials, we need to recall the definition of a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers and coefficients that are real numbers. Let's analyze each function one by one: 1. **\( f(x) = 3 x (x^{4} - 6 x^{3} + 1) \)**: - This is a product of a constant and a polynomial (since \( x^{4} - 6 x^{3} + 1 \) is a polynomial). - Therefore, \( f(x) \) is a polynomial. 2. **\( g(x) = 5 x^{7} + 2 x^{-3} - 7 x + 10 \)**: - This function contains the term \( 2 x^{-3} \), which has a negative exponent. - Therefore, \( g(x) \) is not a polynomial. 3. **\( y = 8 \sqrt{x} + 6 x - 4 \)**: - The term \( 8 \sqrt{x} \) can be rewritten as \( 8 x^{1/2} \), which has a fractional exponent. - Therefore, \( y \) is not a polynomial. 4. **\( h(x) = 5^{x} + 8 x^{3} - 2 x \)**: - The term \( 5^{x} \) is an exponential function, not a polynomial. - Therefore, \( h(x) \) is not a polynomial. 5. **\( f(x) = -3 x^{4} + \frac{5}{2} x \)**: - This function consists of terms with non-negative integer powers of \( x \) (specifically \( x^{4} \) and \( x^{1} \)). - Therefore, this \( f(x) \) is a polynomial. In summary, the functions that are polynomials are: - \( f(x) = 3 x (x^{4} - 6 x^{3} + 1) \) - \( f(x) = -3 x^{4} + \frac{5}{2} x \) Thus, the polynomials are: - \( f(x) \) (first one) - \( f(x) \) (last one)