Questions: Determine whether the following function is a polynomial function. If the function is a polynomial function, identify the degree and the constant term. F(x)=7 x^6 - π x^2 + 1/2 Determine whether F(x) is a polynomial or not. Select the correct choice below and, if necessary, fill in the A. It is not a polynomial because the variable x is raised to the power, which is not a nonnegative (Type an integer or a fraction.) B. It is a polynomial of degree 6. (Type an integer or a fraction.) C. It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomial Write the polynomial in standard form. Then identify the leading term and the constant term. Select the correct A. The polynomial in standard form is F(x)= with the leading term and the constant 7. B. The function is not a polynomial.

Determine whether the following function is a polynomial function. If the function is a polynomial function, identify the degree and the constant term. F(x)=7 x^6 - π x^2 + 1/2 Determine whether F(x) is a polynomial or not. Select the correct choice below and, if necessary, fill in the A. It is not a polynomial because the variable x is raised to the power, which is not a nonnegative (Type an integer or a fraction.) B. It is a polynomial of degree 6. (Type an integer or a fraction.) C. It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomial Write the polynomial in standard form. Then identify the leading term and the constant term. Select the correct A. The polynomial in standard form is F(x)= with the leading term and the constant 7. B. The function is not a polynomial.
Transcript text: Determine whether the following function is a polynomial function. If the function is a polynomial function, identify the degree and the constant term. \[ F(x)=7 x^{6}-\pi x^{2}+\frac{1}{2} \] Determine whether $F(x)$ is a polynomial or not. Select the correct choice below and, if necessary, fill in the A. It is not a polynomial because the variable x is raised to the $\square$ power, which is not a nonnegative (Type an integer or a fraction.) B. It is a polynomial of degree 6. (Type an integer or a fraction.) C. It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomial Write the polynomial in standard form. Then identify the leading term and the constant term. Select the correct A. The polynomial in standard form is $F(x)=$ $\square$ with the leading term $\square$ and the constant $\square$ 7. B. The function is not a polynomial.
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Solution

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Solution Steps

To determine if a function is a polynomial, check if all the exponents of the variable are non-negative integers and if the coefficients are real numbers. For the given function \( F(x) = 7x^6 - \pi x^2 + \frac{1}{2} \), verify these conditions. If it is a polynomial, identify the degree, leading term, and constant term. The degree is the highest power of \( x \), the leading term is the term with the highest power, and the constant term is the term without \( x \).

Step 1: Determine if the Function is a Polynomial

A polynomial function is an expression that consists of variables raised to nonnegative integer powers and coefficients. The given function is:

\[ F(x) = 7x^6 - \pi x^2 + \frac{1}{2} \]

  • The term \(7x^6\) has a variable \(x\) raised to the power of 6, which is a nonnegative integer.
  • The term \(-\pi x^2\) has a variable \(x\) raised to the power of 2, which is also a nonnegative integer. The coefficient \(-\pi\) is a real number, which is acceptable in a polynomial.
  • The term \(\frac{1}{2}\) is a constant term.

Since all terms meet the criteria for a polynomial, \(F(x)\) is indeed a polynomial function.

Step 2: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable \(x\) in the polynomial. In the function \(F(x) = 7x^6 - \pi x^2 + \frac{1}{2}\), the highest power of \(x\) is 6. Therefore, the degree of the polynomial is 6.

Step 3: Write the Polynomial in Standard Form and Identify Leading and Constant Terms

A polynomial is in standard form when its terms are written in descending order of their powers. The given polynomial is already in standard form:

\[ F(x) = 7x^6 - \pi x^2 + \frac{1}{2} \]

  • The leading term is the term with the highest power of \(x\), which is \(7x^6\).
  • The constant term is the term without any \(x\), which is \(\frac{1}{2}\).

Final Answer

  • The function \(F(x)\) is a polynomial of degree 6.
  • The polynomial in standard form is \(F(x) = 7x^6 - \pi x^2 + \frac{1}{2}\).
  • The leading term is \(7x^6\).
  • The constant term is \(\frac{1}{2}\).

\[ \boxed{\text{The function is a polynomial of degree 6.}} \]

\[ \boxed{\text{The polynomial in standard form is } F(x) = 7x^6 - \pi x^2 + \frac{1}{2}.} \]

\[ \boxed{\text{The leading term is } 7x^6 \text{ and the constant term is } \frac{1}{2}.} \]

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