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

A polynomial is an expression that involves a sum of powers in one or more variables multiplied by coefficients, without any negative or fractional powers. So, let's dive into the exciting world of polynomials! First, the function \( f(x)=3 x\left(x^{4}-6 x^{3}+1\right) \) is a polynomial! By distributing that, you'll see it can be expressed as \( 3x^{5} - 18x^{4} + 3x \), which consists of whole-number powers of \( x \). On the flip side, \( g(x)=5 x^{7}+2 x^{-3}-7 x+10 \) is not a polynomial because of the \( 2 x^{-3} \) term—negative exponents just don’t belong in our polynomial party! Similarly, \( y=8 \sqrt{x}+6 x-4 \) and \( h(x)=5^{x}+8 x^{3}-2 x \) are off the guest list too due to their square root and exponential components, respectively. Last but not least, \( f(x)=-3 x^{4}+\frac{5}{2} x \) is indeed a polynomial despite that pesky fraction; it just needs whole number powers! So, let’s recap: \( f(x) \) and \( f(x) = -3 x^{4} + \frac{5}{2} x \) are our polynomial stars! 🎉