Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term. \[ F(x)=3 x^{4}-\pi x^{2}+\frac{1}{6} \] Determine whether $F(x)$ is a polynomial or not. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. It is not a polynomial because the variable $x$ is raised to the $[$ power, which is not a nonnegative integer. (Iype an integer or a fraction) B. It is a polynomial of degree $\square$. (Type an integer or a fraction) C. It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomia in the denominator is of positive degree. Write the polynomial in standard form. Then identify the leading term and the constant term. Select the correct choce and, if necessary, fill in the answer boxes to complete your choice. A. The polynomial in standard form is $F(x)=\square$ with the leading term $\square$ and the constant $\square$ (Use integers or fractions for any numbers in the expressions.) B. The function is not a polynomial.
Solution
Step 1 ：The function given is \(F(x)=3 x^{4}-\pi x^{2}+\frac{1}{6}\). A polynomial function is a function that can be expressed in the form \(p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_2, a_1, a_0\) are constants and \(n\) is a nonnegative integer.
Step 2 ：Looking at the function \(F(x)\), it seems to fit this definition. The highest power of \(x\) is 4, so the degree of the polynomial should be 4.
Step 3 ：The function is already in standard form. The leading term is the term with the highest power of \(x\), which is \(3x^4\). The constant term is the term without \(x\), which is \(\frac{1}{6}\).
Step 4 ：\(\boxed{\text{The function } F(x) \text{ is a polynomial function. The correct choice is B. It is a polynomial of degree 4.}}\)
Step 5 ：\(\boxed{\text{The polynomial in standard form is } F(x)=3 x^{4}-\pi x^{2}+\frac{1}{6} \text{ with the leading term } 3x^4 \text{ and the constant } \frac{1}{6}. \text{ The correct choice is A.}}\)
